Electrokinetic And Colloid Transport Phenomena 9780471788829, 0471788821

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ELECTROKINETIC AND COLLOID TRANSPORT PHENOMENA JACOB H. MASLIYAH University of Alberta

SUBIR BHATTACHARJEE University of Alberta

A JOHN WILEY & SONS, INC., PUBLICATION

Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and authors have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor the authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.

Library of Congress Cataloging-in-Publication Data is available. Electrokinetic and Colloid Transport Phenomena Masliyah, Jacob H. and Bhattacharjee, Subir ISBN 13: 978-0-471-78882-9 ISBN 10: 0-471-78882-1 Printed in the United States of America 10

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This work is dedicated to our parents, Heskel Haim & Naima Masliyah, and Sambhu Nath & Swarna Bhattacharjee. —– For our families, Odette, Tamara, Ruth, & Daniel, and Kamaljit & Arya. —– A tribute to our mentors

CONTENTS

PREFACE

xvii

COPYRIGHT ACKNOWLEDGMENTS

xxi

CHAPTER 1 1.1 1.2 1.3 1.4 1.5 1.6

1.7

2.5

1

Units / 2 Physical Constants and Conversion Factors / 3 Frequently used Functions / 4 Vector Operations / 6 Tensor Operations / 9 Vector and Tensor Integral Theorems / 11 1.6.1 The Divergence and Gradient Theorems / 11 1.6.2 The Stokes Theorem / 12 References / 12

CHAPTER 2 2.1 2.2 2.3 2.4

MATHEMATICAL PRELIMINARIES

COLLOIDAL SYSTEMS

13

The Colloidal State / 13 Colloidal Phenomena / 16 Stabilization of Colloids / 21 Preparation of Colloidal Systems / 23 2.4.1 Dispersion Methods / 23 2.4.2 Condensation Methods / 24 Purification of Sols / 26 vii

viii

CONTENTS

2.6 2.7 2.8 2.9

A Historical Summary / 27 Electrokinetic Phenomena in Modern Colloid Science / 29 Nomenclature / 30 References / 31

CHAPTER 3 3.1

3.2

3.3 3.4 3.5 3.6 3.7 3.8 3.9

4.2

4.3

33

Basic Electrostatics in Free Space / 33 3.1.1 Fundamental Principles of Electrostatics / 33 3.1.2 Electric Field Strength / 36 3.1.3 The Gauss Law / 43 3.1.4 Electric Potential / 44 Summary of Electrostatic Equations in Free Space / 50 3.2.1 Integral Form / 50 3.2.2 Differential Form / 50 Electrostatic Classification of Materials / 51 Basic Electrostatics in Dielectrics / 56 Boundary Conditions for Electrostatic Equations / 62 Maxwell Stress for a Linear Dielectric / 68 Maxwell’s Equations of Electromagnetism / 73 Nomenclature / 74 References / 75

CHAPTER 4 4.1

ELECTROSTATICS

APPLICATION OF ELECTROSTATICS

Two-Dimensional Dielectric Slab in an External Electric Field / 77 4.1.1 Electric Potential and Field Strength / 78 4.1.2 Polarization Surface Charge Density / 80 4.1.3 Maxwell Electrostatic Stress / 81 A Dielectric Sphere in an External Electric Field / 83 4.2.1 Electric Potential and Field Strength / 84 4.2.2 Polarization Surface Charge Density / 86 4.2.3 Maxwell Electrostatic Stress on the Dielectric Sphere / 87 A Conducting Sphere in an External Electric Field / 91 4.3.1 Electric Potential and Field Strength for a Conducting Sphere / 91 4.3.2 Surface Charge Density for a Conducting Sphere / 92 4.3.3 Maxwell Electrostatic Stress on the Conducting Sphere / 94

77

CONTENTS

4.4 4.5 4.6 4.7 4.8

Charged Disc and Two Parallel Discs in a Dielectric Medium / 95 Point Charges in a Dielectric Medium / 97 Nomenclature / 100 Problems / 101 References / 103

CHAPTER 5 5.1

5.2

5.3

5.4

5.5 5.6

5.7

5.8

ix

ELECTRIC DOUBLE LAYER

Electric Double Layers at Charged Interfaces / 105 5.1.1 Origin of Interfacial Charge / 106 5.1.2 Electrical Potential Distribution Near an Interface / 108 5.1.3 The Boltzmann Distribution / 109 Potential for Planar Electric Double Layer / 111 5.2.1 Gouy–Chapman Analysis / 111 5.2.2 Debye–Hückel Approximation / 116 5.2.3 Surface Charge Density / 122 5.2.4 Ionic Concentrations in Electric Double Layers / 125 5.2.5 High Surface Potentials and Counterion Analysis / 128 Potential for Curved Electric Double Layer / 130 5.3.1 Spherical Geometry: Debye–Hückel Approximation / 130 5.3.2 Cylindrical Geometry: Debye–Hückel Approximation / 136 Electrostatic Interaction between Two Planar Surfaces / 138 5.4.1 Force between Two Charged Planar Surfaces / 138 5.4.2 Surface Charge Density for Planar Surfaces: Overlapping Double Layers / 147 Electrostatic Potential Energy / 152 Electrostatic Interactions between Curved Geometries / 155 5.6.1 The Derjaguin Approximation / 157 5.6.2 Linear Superposition Approximation / 162 5.6.3 Other Approximate Solutions / 164 Models of Surface Potentials / 165 5.7.1 Indifferent Electrolytes / 166 5.7.2 Ionizable Surfaces / 167 Zeta Potential / 169

105

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CONTENTS

5.9

5.10 5.11 5.12

Summary of Gouy–Chapman Model / 171 5.9.1 Arbitrary Electrolyte / 171 5.9.2 Symmetrical (z : z) Electrolyte / 172 5.9.3 Forms of Various Notations / 173 Nomenclature / 173 Problems / 175 References / 176

CHAPTER 6 6.1 6.2

6.3 6.4 6.5 6.6 6.7

ELECTROKINETIC PHENOMENA

221

Electroosmosis / 221 Streaming Potential / 222 Electrophoresis / 222 Sedimentation Potential / 223 Non-Equilibrium Processes and Onsager Relationships / 223 Nomenclature / 226 References / 226

CHAPTER 8 8.1 8.2

179

Single-Component System / 180 Multicomponent Systems / 181 6.2.1 Basic Definitions / 181 6.2.2 Mass Conservation / 183 6.2.3 Convection–Diffusion–Migration Equation / 185 6.2.4 Current Density / 191 6.2.5 Conservation of Charge / 198 6.2.6 Binary Electrolyte Solution / 199 6.2.7 Boltzmann Distribution / 201 6.2.8 Momentum Equations / 203 Hydrodynamics of Colloidal Systems / 205 Summary of Governing Equations / 212 Nomenclature / 217 Problems / 219 References / 219

CHAPTER 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7

FUNDAMENTAL TRANSPORT EQUATIONS

FLOW IN MICROCHANNELS

Liquid Flow in Channels / 229 Electroosmotic Flow in a Slit Charged Microchannel / 230 8.2.1 Electric Potential / 230 8.2.2 Flow Velocity / 235 8.2.3 Volumetric Flow Rate / 238

229

CONTENTS

8.3 8.4 8.5 8.6 8.7 8.8

8.9 8.10 8.11

8.12 8.13 8.14

Electroosmotic Flow in a Closed Slit Microchannel / 240 Effectiveness of Electroosmotic Flow / 243 Electric Current in Electroosmotic Flow in Slit Channels / 244 Streaming Potential in Slit Channels / 251 Electroviscous Flow in Slit Microchannels / 252 Electroosmotic flow in a Circular Charged Capillary / 253 8.8.1 Thin Double Layers: Helmholtz–Smoluchowski Equation, κa ≫ 1 / 257 8.8.2 Thick Double Layers, κa ≪ 1 / 257 8.8.3 Current Flow in Electroosmosis / 262 8.8.4 Streaming Potential Analysis / 265 8.8.5 Electroviscous Effect / 266 High Surface Potential / 268 Surface Conductance / 270 Solute Dispersion in Microchannels / 274 8.11.1 Diffusional and Hydrodynamic Dispersion / 275 8.11.2 Convective-Diffusional Transport Through Channels / 278 8.11.3 Dispersion in a Slit Microchannel / 283 Nomenclature / 286 Problems / 287 References / 290

CHAPTER 9 9.1 9.2

9.3

9.4

xi

ELECTROPHORESIS

Introduction / 295 Electrophoresis of a Single Charged Sphere / 296 9.2.1 Transport Mechanisms in Electrophoresis / 296 9.2.2 General Governing Equations / 298 9.2.3 Boundary Conditions / 299 9.2.4 Electrophoresis in the Limit κa ≪ 1 / 303 9.2.5 Electrophoresis in the Limit κa ≫ 1 / 306 Improved Solutions: Arbitrary Debye Length / 308 9.3.1 Perturbation Approach / 309 9.3.2 Henry’s Solution / 311 9.3.3 Effect of Particle Conductivity and Shape / 319 9.3.4 Alternate Forms of the Electrophoretic Velocities / 322 9.3.5 Solutions Accounting for Relaxation Effects / 324 Electrophoretic Mobility in Concentrated Suspensions / 327 9.4.1 Cell Models for the Hydrodynamic Problem / 328 9.4.2 The Levine–Neale Cell Model / 333

295

xii

CONTENTS

9.5 9.6 9.7 9.8

9.4.3 The Ohshima Cell Model / 340 9.4.4 Suspension Electric Conductivity / 344 9.4.5 The Shilov–Zharkikh Cell Model / 346 9.4.6 Accuracy of the Cell Model Predictions / 352 Circular Cylinders Normal to the Electric Field / 354 Nomenclature / 356 Problems / 358 References / 359

CHAPTER 10 SEDIMENTATION POTENTIAL 10.1 10.2 10.3

10.4 10.5 10.6 10.7

Sedimentation of Uncharged Spherical Particles / 363 Concept of Sedimentation Potential and Velocity / 365 Dilute Suspensions: Ohshima’s Model / 370 10.3.1 Fundamental Governing Equations / 370 10.3.2 Boundary Conditions / 373 10.3.3 Perturbation Approach / 374 10.3.4 Sedimentation Velocity: Single Charged Sphere / 376 10.3.5 Sedimentation Potential: Dilute Suspensions / 378 Sedimentation Potential of Concentrated Suspensions / 381 Nomenclature / 386 Problems / 388 References / 389

CHAPTER 11 LONDON–VAN DER WAALS FORCES AND THE DLVO THEORY 11.1 11.2

363

Dispersion Forces Between Bodies in Vacuum / 391 Hamaker’s Approach / 393 11.2.1 Approximate Expressions for van der Waals Interaction / 398 11.2.2 Cohesive Work and Hamaker’s Constant / 401 11.2.3 Electromagnetic Retardation / 402 11.3 Effects of Intervening Medium / 403 11.4 DLVO Theory of Colloidal Interactions / 406 11.5 Schulze–Hardy Rule / 409 11.6 Verification of the DLVO Theory / 412 11.7 Limitations of DLVO Theory / 415 11.7.1 Major Assumptions in DLVO Model of Colloidal Interactions / 416 11.7.2 When DLVO Theory Falls Short / 418

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CONTENTS

xiii

11.8 Nomenclature / 420 11.9 Problems / 421 11.10 References / 422 CHAPTER 12 COAGULATION OF PARTICLES 12.1 12.2 12.3 12.4 12.5

12.6

12.7 12.8 12.9

Introduction / 427 Dynamics of Coagulation / 428 Brownian Motion / 429 Collision Frequency / 432 Brownian Coagulation / 434 12.5.1 The Smoluchowski Solution without a Field Force / 434 12.5.2 Effect of a Field Force / 437 Coagulation due to Shear / 448 12.6.1 The Smoluchowski Solution in the Absence of Brownian Motion / 448 12.6.2 Coagulation due to Shear in the Absence of Brownian Motion: With Hydrodynamic and Field Forces / 451 Nomenclature / 462 Problems / 464 References / 466

CHAPTER 13 DEPOSITION OF COLLOIDAL PARTICLES 13.1 13.2

13.3

13.4

13.5

427

Introduction / 469 Classical Deposition Mechanisms / 471 13.2.1 Brownian Diffusion: Classical Convection–Diffusion Transport / 471 13.2.2 Interception Deposition / 471 13.2.3 Inertial Deposition / 475 Eulerian Approach / 477 13.3.1 Deposition Due to Brownian Diffusion Without External Forces: Spherical Collector / 478 13.3.2 Deposition due to Brownian Diffusion with External Forces: Stagnation Flow / 482 13.3.3 Deposition due to Brownian Diffusion with External Forces: Spherical Collectors / 490 Lagrangian Approach / 497 13.4.1 Particle Collisions on a Spherical Collector: With the Presence of External Forces / 497 Deposition Efficiency and Sherwood Number / 509

469

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CONTENTS

13.6 Experimental Verifications / 512 13.7 Application of Deposition Theory / 521 13.7.1 Deposition in Porous Media / 521 13.7.2 Colloid Transport Models in Porous Media / 524 13.8 Summary of Dimensionless Groups / 527 13.8.1 Dimensionless Groups in the Flux Equation / 527 13.8.2 Dimensionless Groups in the Trajectory Equation / 527 13.9 Nomenclature / 528 13.10 Problems / 531 13.11 References / 533 CHAPTER 14 NUMERICAL SIMULATION OF ELECTROKINETIC PHENOMENA 14.1 14.2

14.3

14.4

14.5 14.6

Tools and Methods for Computer Based Simulations / 538 Numerical Solution of the Poisson–Boltzmann Equation / 541 14.2.1 Problem Formulation / 543 14.2.2 Finite Element Formulation / 546 14.2.3 Mesh Generation / 547 14.2.4 Solution Methodology / 549 14.2.5 Postprocessing: Calculation of the EDL Force / 550 14.2.6 Validation of Numerical Results / 553 14.2.7 EDL Interaction Force on Particles in a Charged Capillary / 557 Flow of Electrolyte in a Charged Cylindrical Capillary / 559 14.3.1 Problem Formulation / 562 14.3.2 Mesh Generation and Numerical Solution / 567 14.3.3 Case 1: Streaming Potential Across a Capillary Microchannel / 570 14.3.4 Case 2: Transient Analysis of Electrolyte Transport in a Capillary Microchannel / 577 14.3.5 Case 3: Electroosmotic Flow due to Axial Pressure and Electric Potential Gradients / 581 Analysis of Electrophoretic Mobility / 587 14.4.1 Problem Formulation / 590 14.4.2 Mesh Generation and Numerical Solution / 598 14.4.3 Representative Simulation Results / 602 Concluding Remarks / 605 Nomenclature / 606

537

CONTENTS

14.7 14.8

xv

Problems / 608 References / 608

CHAPTER 15 ELECTROKINETIC APPLICATIONS

613

15.1 15.2

Introduction / 613 Electrokinetic Salt Rejection in Porous Media and Membranes / 613 15.3 Electroosmotic Control of Hazardous Wastes / 617 15.4 Iontophoretic Delivery of Drugs / 619 15.5 Flotation of Oil Droplets and Fine Particles / 622 15.6 Rheology of Colloidal Suspensions / 625 15.6.1 Historical Background / 625 15.6.2 Hard Sphere Model / 626 15.6.3 Electroviscous Effects / 628 15.7 Bitumen Extraction From Oil Sands / 632 15.7.1 Zeta Potential of Oil Sand Components / 635 15.7.2 Zeta Potential Distribution Measurements Technique / 639 15.7.3 Atomic Force Microscope Technique / 640 15.7.4 Electrokinetic Phenomena in Bitumen Recovery from Oil Sands / 641 15.8 Microfluidic and Nanofluidic Applications / 650 15.8.1 Measurement of Zeta Potential of Macroscopic Surfaces / 653 15.8.2 AC Electrokinetics: Application in Membrane Filtration / 659 15.9 Nomenclature / 666 15.10 References / 668

INDEX

673

PREFACE

Electrokinetics is a subject that has been at the core of numerous fundamental advancements in the field of colloid science for over a century. Electrokinetics is a self-contained body of science that has led to spectacular applications in separations, characterization of surface properties, manipulation of colloidal materials, and facilitation of fluid transport in microchannels. For instance, electrophoresis is one of the common techniques for separation of biological macromolecules (such as proteins). Reverse osmosis is nowadays almost a household name in the context of water filtration and purification. Streaming potential or electrophoretic mobility measurements are extremely common for evaluation of the surface potentials of charged surfaces or colloidal particles in a variety of industrial applications. The above examples provide a few glimpses of the diverse applications of electrokinetic transport phenomena. Concurrently, the theoretical knowledge base underlying electrokinetics has undergone significant improvement over the past several decades, providing a complete and unified picture of the basic physical and chemical phenomena associated with these transport processes. The subject of electrokinetics is also fraught with considerable enigma and misconceptions, perhaps leading to a general apathy toward it by a large body of scientists and researchers. There are even notions that the theoretical treatment in this area is often incomplete, and sometimes inaccurate. A plausible reason behind this ambivalent approach to this subject is the lack of a coordinated understanding of fluid dynamics, mass transport, and electrostatics—the three pillars on which the subject stands. Very rarely do we provide students in a single discipline of science and engineering a complete course covering all three aspects of electrokinetic transport in sufficient detail. Often such lack of coordination has indeed led to some misconceptions and inaccurate models. However, it is perhaps unfair to consider this highly xvii

xviii

PREFACE

evolved and extraordinarily beautiful subject incomplete. There has been a resurgence of interest in this fairly mature subject since the advent of microfluidics during the last decade. In light of this renewed interest in this subject, it is perhaps pertinent to systematically reevaluate the theoretical underpinnings of electrokinetic transport phenomena. The present book can be considered as an outgrowth of an earlier book titled Electrokinetic Transport Phenomena, published in 1994. The guiding principle of the earlier book was to provide a detailed description of the fundamental electrochemical transport processes underlying electrokinetic and colloid transport phenomena. Particular emphasis was laid toward rendering the mathematical developments sufficiently tractable such that a graduate or a senior undergraduate student approaching the subject for the first time can easily interpret the theories. We follow this style in the present book as well. The present book aims to provide a fundamental perspective of electrokinetic and colloid transport processes, emphasizing the coupling between electrical, mass transport, and fluid dynamical aspects of these complex transport phenomena. In the first five chapters, the principles of electrostatics are described, providing a comprehensive coverage of basic electrostatics in vacuo and perfect dielectrics, as well as electric double layer phenomena in electrolyte solutions. Following this, Chapter six elaborates the fundamental aspects of mass transport phenomena in electrolyte systems. Chapters seven through ten describe the four basic electrokinetic phenomena, namely, electroosmosis, streaming potential, electrophoresis, and sedimentation potential. In Chapter eleven, we present the Derjaguin–Landau–Verwey–Overbeek (DLVO) model of colloidal interactions, which forms the basis of subsequent discussion on two aspects of colloid transport phenomena, namely, coagulation and particle deposition, in Chapters twelve and thirteen, respectively. Chapter fourteen presents a self-contained description of numerical approaches for solving electrokinetic transport problems. Finally, in Chapter fifteen, we provide several examples of application of electrokinetic and colloid transport phenomena in different disciplines of science and engineering. The book has been a product of several years of teaching graduate and senior undergraduate students in the subjects of colloidal phenomena, colloid transport, electrokinetics, and microfluidics. The courses were offered independently by both of the authors at various institutions in Chemical Engineering, Materials Engineering, and Civil and Environmental Engineering, as well as Mechanical Engineering. The audience for these courses virtually always consisted of a diverse array of graduate students or senior undergraduate students from different branches of engineering (Chemical, Civil, Environmental, Electrical, and Mechanical), physics, and chemistry. The courses took titles ranging from Colloidal Phenomena, Colloidal Phenomena in Aquatic Systems, Colloidal Hydrodynamics, Colloidal and Electrokinetic Phenomena, to Microfluidics and Electrokinetics. The subject of electrokinetics is generally regarded as highly mathematical in nature. Most treatments of this subject rely extensively on vector and tensor notations. Experience gleaned from our teaching this course leads us to believe that using concise vector and tensor notations often makes the subject appear fairly complex to a graduate

PREFACE

xix

student being initiated to this area. Consequently, our approach has been to use the “exploded” forms of the governing equations. This might initially give the reader the impression that the book contains an unusually large number of equations and formulas. However, our teaching experience shows that the detailed derivations of the mathematical models (where we rarely skimp on intermediate steps) can make the process of learning and applying the mathematical principles fairly simple for a student with a modest mathematical background. The style of the mathematics used in the book should also render writing computer codes on the basis of these formulae fairly straightforward. Although the primary objective of the book is to serve as a graduate-level textbook, we feel that it can also serve as a comprehensive and updated reference material on the theoretical aspects of electrokinetic and colloid transport phenomena relevant to a wide range of practicing engineers and scientists. The writing of the book led us through a significant learning process. In this respect, we are grateful to Dr. Emiliy Zholkovskij for clarifying numerous questions, doubts, and misconceptions, and generally illuminating us with a clear and unambiguous theoretical and physical picture of the subject matter. His contributions toward writing the material on hydrodynamic dispersion in Chapter 8 are gratefully acknowledged. Finally, without his meticulous proofreading of the mathematical formulae and the derivations, the book would have been incomplete. We are grateful to Mr. Glen Thomas for his assistance with the proofreading of the manuscript. We would also like to acknowledge many of our graduate student researchers, whose contributions are featured in many parts of the book. The financial support of Natural Sciences and Engineering Research Council (NSERC), Canada Research Chairs program (CRC), Canada Foundation for Innovation (CFI), Alberta Science and Research Authority, Alberta Energy Research Institute, Alberta Ingenuity Fund, and several other funding sources toward our research has been the catalyst that encouraged the learning process leading to this book. Our sincere thanks to Dr. Arza Seidel from John Wiley & Sons for her patience in dealing with our interminable revisions and her generous help with the final production of the manuscript. Finally, we are thankful to our families and colleagues for understanding our preoccupation with this book, and foregoing their demands on our time. We hope that this book serves its intended purpose. We will be rewarded if the material helps make the subject of electrokinetics readily accessible to a broad range of scientific professionals. Finally, virtually no work that is human could be complete and error-free. Despite our careful revisions, there may be errors and omissions that have crept into the manuscript. We will be grateful to be informed of any errors and omissions in this work so that we can improve the book in subsequent editions. JACOB H. MASLIYAH SUBIR BHATTACHARJEE Edmonton, Alberta August, 2005

COPYRIGHT ACKNOWLEDGMENTS

Table 2.1: Reproduced by permission of the Royal Society of Chemistry. Table 2.2: Copyright (1989), with permission from Cambridge University Press. Table 3.4: Reprinted from Intermolecular and Surface Forces, Israelachvili, J.N., Copyright (1985), with permission from Elsevier. Table 6.2: Copyright (2002) from CRC Handbook of Chemistry and Physics by Lide, D.R. (Ed.). Reproduced by permission of Routledge/Taylor & Francis Group, LLC. Table 6.4: Reprinted from Atkinson, G., Electrochemical Information, in American Institute of Physics Handbook, 3rd ed., Gray, D.E. (Ed.), Copyright (1972), with permission from The McGraw-Hill Companies. Figure 8.20: Copyright (1975), with permission from Elsevier. Figure 9.10: Reproduced by permission of The Royal Society of Chemistry. Figure 9.11: Reproduced by permission of The Royal Society of Chemistry. Figure 9.17: Copyright (1989), with permission from Elsevier. Figure 9.18: Copyright (1989), with permission from Elsevier. Figure 9.19: Copyright (1989), with permission from Elsevier. Table 11.5: Copyright (1989), with permission from Cambridge University Press. Figure 11.6: Reprinted from Introduction to Colloid and Surface Chemistry, Shaw, D.J., Copyright (1980), with permission from Elsevier. xxi

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COPYRIGHT ACKNOWLEDGMENTS

Figure 11.7: Copyright (1989), with permission from Cambridge University Press. Figure 11.8: Reprinted from Intermolecular and Surface Forces, Israelachvili, J.N., Copyright (1985), with permission from Elsevier. Figure 11.9: Copyright (1978), with permission from The Royal Society of Chemistry. Figure 11.10: Copyright (1981), with permission from Elsevier. Figure 12.6: Copyright (1989), with permission from Cambridge University Press. Table 12.2: Copyright (1989), with permission from Cambridge University Press. Table 12.3: Reproduced with permission. Copyright ©1977 AIChE. Figure 12.12: Reprinted from Colloidal Hydrodynamics, van de Ven, T.G.M., Copyright (1989), with permission from Elsevier. Figure 12.13: Copyright (1983), with permission from Cambridge University Press. Figure 12.14: Copyright (1981), with permission from Elsevier. Figure 12.17: Copyright (1983), with permission from Cambridge University Press. Figure 12.18: Reproduced with permission. Copyright ©1977 AIChE. Figure 12.20: Copyright (1985), with permission from Elsevier. Figure 13.5: Copyright (1969), with permission from Cambridge University Press. Table 13.1: Copyright (1973), with permission from Elsevier. Figure 13.9: Copyright (1983), with kind permission of Springer Science and Business Media. Figure 13.10: Copyright (1983), with kind permission of Springer Science and Business Media. Figure 13.11: Reproduced with permission. Copyright ©1974 AIChE. Figure 13.12: Reproduced with permission. Copyright ©1974 AIChE. Figure 13.13: Reproduced with permission. Copyright ©1974 AIChE. Figure 13.14: Reproduced with permission. Copyright ©1974 AIChE. Figure 13.15: Reproduced with permission. Copyright ©1974 AIChE. Figure 13.16: Reproduced with permission. Copyright ©1974 AIChE. Figure 13.20: Copyright (1973), with permission from Elsevier. Figure 13.21: Copyright (1973), with permission from Elsevier. Figure 13.23: Copyright (1973), with permission from Elsevier. Figure 13.24: Copyright (1973), with permission from Elsevier.

COPYRIGHT ACKNOWLEDGMENTS

xxiii

Figure 13.25: Copyright (1973), with permission from Elsevier. Figure 13.26: Copyright (1973), with permission from Elsevier. Figure 13.31: Copyright (1973), with permission from Elsevier. Figure 13.32: Copyright (1973), with permission from Elsevier. Figure 13.33: Copyright (1973), with permission from Elsevier. Figure 14.4: Reprinted with permission. Copyright (2003) American Chemical Society. Figure 14.6: Reprinted with permission. Copyright (2003) American Chemical Society. Figure 14.15: Copyright (2005), with permission from Elsevier. Figure 15.1: Reprinted with permission. Copyright (1972) American Chemical Society. Figure 15.2: Reprinted with permission. Copyright (1972) American Chemical Society. Figure 15.3: Copyright (1987), with permission from Elsevier. Figure 15.6: Copyright (1988), with permission from Elsevier. Figure 15.7: Copyright (1988), with permission from Elsevier. Figure 15.12: Copyright (1972), with permission from Elsevier. Figure 15.14: Reproduced by permission of The Royal Society of Chemistry. Figure 15.15: Copyright (1972), with permission from Elsevier. Figure 15.18: Copyright (1992), with permission from Elsevier. Figure 15.22: Reprinted with permission. Copyright (2003) American Chemical Society. Figure 15.23: Reprinted with permission. Copyright (2003) American Chemical Society. Figure 15.25: Reproduced with permission. Copyright ©2004 AIChE. Figure 15.26: Copyright (2002), with permission from Elsevier. Figure 15.30: Reprinted with permission. Copyright (2003) American Chemical Society. Figure 15.31: Reprinted with permission. Copyright (2003) American Chemical Society. Figure 15.32: Reprinted with permission. Copyright (2003) American Chemical Society.

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COPYRIGHT ACKNOWLEDGMENTS

Figure 15.33: Reprinted with permission. Copyright (2003) American Chemical Society. Figure 15.35: Copyright (2005), with permission from Elsevier. Figure 15.36: Copyright (2005), with permission from Elsevier. Figure 15.37: Copyright (2005), with permission from Elsevier. Figure 15.38: Copyright (2005), with permission from Elsevier. Figure 15.45: Copyright (2005), with permission from Elsevier.

CHAPTER 1

MATHEMATICAL PRELIMINARIES

Treatment of electrokinetic transport phenomena requires understanding of fluid mechanics, colloidal phenomena, and the interaction of charged particles, surfaces, and electrolytes with an external electrical field. Accordingly, dealing with electrokinetic transport processes requires familiarity with the units and dimensions of fundamental quantities from a diverse range of subjects. In this chapter, we outline the pertinent units and dimensions of the fundamental quantities encountered in electrokinetic transport processes. Historically, the centimeter-gram-second (cgs) system of units was widely used in most colloid science and electrokinetics literature. However, with the popularity of the Système Internationale d’Unités (the SI system), most of the modern treatment of these subjects are based on SI units. Accordingly, most of the topics covered in this book are based on the SI system. To facilitate the conversion of other units into the SI system, the first few sections of this Chapter are devoted to definitions of the fundamental units and dimensions, description of the derived units in the SI system, values of the commonly encountered physical constants in various units, and conversion factors for different quantities from SI to non-SI units. The latter half of the Chapter outlines some of the mathematical fundamentals required to develop the theoretical treatments in the rest of the book, including a short primer of series functions, vector, and tensor operations.

Electrokinetic and Colloid Transport Phenomena, by Jacob H. Masliyah and Subir Bhattacharjee Copyright © 2006 John Wiley & Sons, Inc.

1

2

MATHEMATICAL PRELIMINARIES

1.1 UNITS The fundamental quantities required in electrokinetic transport analysis are shown in Table 1.1, along with their SI units and symbols. These fundamental quantities can be combined to yield different derived quantities, the units of which are combinations of the fundamental units. Table 1.2 provides some of the commonly used derived quantities and their SI units. To provide a facile transition of the basic dimensions over large ranges, it is often convenient to use scale factors for the basic units. This is particularly important in terms of the length scales used to define the dimensions of extremely small colloidal particles. For instance, it is convenient to express particle sizes in terms of nanometer (nm) or micrometer (µm) instead of meter (m). Similarly, the colloidal forces are conveniently expressed in terms of nano-newtons (nN) or pico-newtons (pN) rather than newtons (N). Table 1.3 provides the commonly used scale factors for the basic units.

TABLE 1.1. Fundamental Quantities Used in Electrokinetic Transport Analysis, their SI Units and Symbols. Quantity

Name of SI Unit

Mass Length Time Temperature Quantity of mass Electric current

kilogram meter second Kelvin mole Ampere

Symbol kg m s K mol A

TABLE 1.2. Derived Quantities and their SI Units. Quantity

SI Unit Name

Force Pressure Energy Power Electric charge Electric potential Electric resistance Electric conductance Electric capacitance Frequency Magnetic inductance Dynamic viscosity Material density

Newton Pascal Joule Watt Coulomb Volt Ohm Siemens Farad Hertz Henry

Symbol N Pa J W C V
S F Hz H

Source: Adapted from Russel et al. (1989) and Probstein (2003).

Definition kg m s−2 N m−2 = kg m−1 s−2 N m = kg m2 s−2 J s−1 = kg m2 s−3 As J C−1 = kg m2 s−3A−1 V A−1 = kg m2 s−3A−2 A V−1 = kg−1 m−2 s3A2 C V−1 = kg−1 m−2 s4A2 s−1 J A−2 = kg m2 s−2A−2 Pa s = N s m−2 = kg m−1 s−1 kg m−3

1.2 PHYSICAL CONSTANTS AND CONVERSION FACTORS

3

TABLE 1.3. Scale Factors for the Basic Units. Factor

Prefix

Symbol

10 10−2 10−3 10−6 10−9 10−12 10−15 10−18

deci centi milli micro nano pico femto atto

d c m µ n p f a

−1

Factor

Prefix

Symbol

10 102 103 106 109 1012 1015 1018

deca hecto kilo mega giga tera peta exa

da h k M G T P E

1.2 PHYSICAL CONSTANTS AND CONVERSION FACTORS The commonly used physical constants and their values in SI units are listed in Table 1.4. The use of non-SI units for various quantities is still common in electrokinetics literature. Conversion factors between SI and other units are provided for some of these quantities in Table 1.5.

TABLE 1.4. Common Physical Constants and their Values in SI Units (Lide, 2001). Quantity Avogadro number Boltzmann constant Elementary charge Faraday constant Magnetic permeability of vacuum Universal gas constant Permittivity of vacuum Planck constant Speed of light in vacuum Standard gravitational acceleration Standard atmospheric pressure (at sea level and 288.16 K) Zero of Celsius scale 1 liter kB T /e at 298.16 K 1 molar solution

Symbol

Value 23

SI Units

NA kB e F µ0

6.022 × 10 1.381 × 10−23 1.602 × 10−19 9.649 × 104 1.2566 × 10−7

R ǫ0

8.314 8.854 × 10−12

h c g

6.626 × 10−34 2.9979 × 108 9.8066

p0

1.01325 × 105

Pa

T0 L

273.15 1.0000028 × 10−3 25.69 × 10−3 1.0

K m3 V mol/dm3 or kmol/m3

M

mol J K−1 C C mol−1 NA−2 , NC−2 s−2 , or Hm−1 JK−1 mol−1 CV−1 m−1 , C2 N−1 m−2 , or Fm−1 Js m s−1 m s−2 −1

Source: Adapted from Hiemenz (1986), Russel et al. (1989), and Probstein (2003).

4

MATHEMATICAL PRELIMINARIES

TABLE 1.5. Conversion Factors for Non-SI Units. Unit

Abbreviation

atmosphere torr atomic mass unit bar electron volt poise liter Angstrom Debye calorie inch pound

atm torr amu bar eV P L Å D cal in lbm

Value 101325 Pa (definition) 133.322 Pa = 1/760 atm 1.6605 × 10−27 kg 1 × 105 Pa 1.6022 × 10−19 J 0.1 kg m−1 s−1 1 × 10−3 m3 = 1 dm3 1 × 10−10 m 3.3356 × 10−30 C m 4.184 J (definition) 0.0254 m (definition) 0.4536 kg

1.3 FREQUENTLY USED FUNCTIONS Here we list some of the common series expansions and functions used frequently in this book. Excellent compilation of mathematical formulae is given by Jeffrey (1995). x2 x3 + + · · · −∞ < x < ∞ 2! 3! ∞
x3 x4 x2 xk ln(1 + x) = x − + − + ··· = −1 < x ≤ 1 (−1)k+1 2 3 4 k k=1 exp(x) = 1 + x +

x2 x3 x4 ln(1 − x) = − x + + + + ··· 2 3 4 



=−

∞
xk k=1

k

−1 ≤ x < 1

1 x3 x5 x7 [exp(x) − exp(−x)] = x + + + + · · · −∞ < x < ∞ 2 3! 5! 7! x2 x4 x6 1 + + + · · · −∞ < x < ∞ cosh(x) = [exp(x) + exp(−x)] = 1 + 2 2! 4! 6! x3 2x 5 17x 7 sinh(x) =x− + − + · · · |x| < π/2 tanh(x) = cosh(x) 3 15 315 sinh(x) =

cosh2 (x) = 1 + sinh2 (x) ∼ cosh(x) x → ∞ sinh(x) =

sinh(x/2) = ±



tanh(x) −→ 1

x→∞

cosh(x) − 1 2

[+ if x > 0 and − if x < 0]



1.3 FREQUENTLY USED FUNCTIONS

5

sinh(x) = 2 sinh(x/2) cosh(x/2) 1 [cosh(2x) − 1] 2 sinh(2x) = 2 sinh(x) cosh(x)   1 + cosh(x) 1/2 cosh(x/2) = 2 sinh2 (x) =

cosh(x) = cosh2 (x/2) + sinh2 (x/2) 1 [1 + cosh(2x)] 2 cosh(2x) = 2 cosh2 (x) − 1

cosh2 (x) =

tanh(x/2) =

sinh(x) cosh(x) − 1 1 − exp(−x) = = 1 + cosh(x) sinh(x) 1 + exp(−x) 1 − tanh(x/2) =

2 exp(−x) 1 + exp(−x)

2 tanh(x/2) 2 coth(x/2) = 1 + tanh2 (x/2) csch2 (x/2) + 2 cosh(2x) − 1 tanh2 (x) = 1 + cosh(2x) 2 tanh(x) tanh(2x) = 1 + tanh2 (x)

tanh(x) =

Some commonly used integrals are provided next. These being indefinite integrals, one should remember to add an integration constant to each result.  dx = −coth(x) sinh2 (x)  dx = tanh(x) cosh2 (x)  dx = tanh(x/2) 1 + cosh(x)  dx = coth(x/2) 1 − cosh(x)  1 tanh(kx)dx = ln[cosh(kx)] k  1 tanh2 (kx)dx = x − tanh(kx) k  1 coth(kx)dx = ln |sinh(kx)| k

6

MATHEMATICAL PRELIMINARIES

1.4 VECTOR OPERATIONS A scalar quantity is defined by a single real number. Temperature and mass are good examples of scalar quantities. A vector quantity is defined by a magnitude and a direction. Velocity of a projectile is a vector quantity. The magnitude of a vector u is given by |u| or simply u. Addition and subtraction of two vectors, u and v, are illustrated in Figure 1.1. Multiplication of a vector u by a scalar quantity s results in changing the magnitude of the vector to s|u| or simply su. The vector direction remains same. A vector can be multiplied with another vector in several ways. Scalar or dot product of two vectors, u and v, is given by u · v = v · u = uv cos φ where φ is the angle formed between the vectors u and v. Here, u and v are the magnitudes of the vectors u and v, respectively. The scalar product rules are Commutative: u·v =v·u Not associative: (u · v) w = u (v · w) Distributive: u · (v + w) = u · v + u · w Vector product or cross product of two vectors u and v is given by another vector defined by u × v = uv sin φ n where φ is the angle between the two vectors and n is a vector of unit length (magnitude) normal to both the vectors u and v in the sense in which a right-handed screw

u– v

u

+v

v

u

u

(a)

(b)

v

Figure 1.1. (a) Addition and (b) subtraction of two vectors.

1.4 VECTOR OPERATIONS

7

would advance if rotated from u to v. A convenient form of a cross product is given by 

i1 u × v =  u1 v1

i2 u2 v2

 i3 u3  v3

where u = u1 i1 + u2 i2 + u3 i3 and v = v1 i1 + v2 i2 + v3 i3 . Here, i1 , i2 , and i3 are orthogonal unit vectors, with u1 , u2 , and u3 being the magnitudes of vector u in the i1 , i2 , and i3 directions, respectively. Similarly, v1 , v2 , and v3 are the magnitudes of the vector v in the i1 , i2 , and i3 directions, respectively. The magnitude of a vector u is given by

 3 
|u| = u = u2 i

i=1

One can also multiply two vectors to obtain a tensor or a dyadic product. The dyadic product of two vectors u and v is given by uv. We will discuss dyadic products in the next section. A compilation of useful vector identities and vector operations is given by Bird et al. (2002). Some commonly used differential vector operations in different orthogonal coordinate systems are given below. Here, ψ is used for a scalar and u is used for a vector.

Cartesian Coordinates (x, y, z) The orthogonal curvilinear coordinates, in the case of Cartesian coordinates, are defined by the unit vectors ix , iy , and iz directed along the x, y, and z coordinates, respectively, and the vector u is given by u = ux ix + uy iy + uz iz The differential operator ∇ is given by ∇=

∂ ∂ ∂ ix + iy + iz ∂x ∂y ∂z

Some useful differential operations are given by ∇ψ =

∂ψ ∂ψ ∂ψ ix + iy + iz ∂x ∂y ∂z

∂ 2ψ ∂ 2ψ ∂ 2ψ + + ∂x 2 ∂y 2 ∂z2 ∂uy ∂uz ∂ux + + ∇ ·u= ∂x ∂y ∂z ∇ 2ψ =

8

MATHEMATICAL PRELIMINARIES

 ix iy iz ∇ × u = ∂/∂x ∂/∂y ∂/∂z ux uy uz       ∂uy ∂uy ∂ux ∂uz ∂uz ∂ux − − − = ix + iy + iz ∂y ∂z ∂z ∂x ∂x ∂y 

Normally, ∇ψ is referred to as the gradient of the scalar ψ, ∇ · u is known as the divergence of the vector u, and ∇ × u is known as the curl of vector u. Here, ∇ 2 is a Laplacian operator. Cylindrical Coordinates (r, θ, z) The cylindrical orthogonal coordinate system is defined by the three orthogonal unit vectors given by ir , iθ , and iz acting along the r, θ, and z directions, respectively. A vector u is given by u = ur ir + uθ iθ + uz iz The differential operator ∇ is given by ∇=

1 ∂ ∂ ∂ ir + iθ + iz ∂r r ∂θ ∂z

Some useful differential operations are given by ∇ψ =

1 ∂ψ ∂ψ ∂ψ ir + iθ + iz ∂r r ∂θ ∂z

∂ 2ψ 1 ∂ψ 1 ∂ 2ψ ∂ 2ψ + + 2 2 + 2 ∂r r ∂r r ∂θ ∂z2 1 ∂ ∂ 1 ∂ (rur ) + ∇ ·u = uθ + uz r ∂r r ∂θ ∂z ∇ 2ψ =

 i 1 r ∂/∂r ∇ ×u= r ur  1 ∂uz = − r ∂θ

 r iθ iz ∂/∂θ ∂/∂z ruθ uz      ∂uθ ∂uz 1 ∂ur ∂ur 1 ∂ − (ruθ ) − ir + iθ + iz ∂z ∂z ∂r r ∂r r ∂θ

Spherical Coordinates (r, θ, φ) The spherical orthogonal coordinate system is defined by the three orthogonal unit vectors given by ir , iθ , and iφ acting along the r, θ, and φ directions, respectively. A vector u is given by u = ur ir + uθ iθ + uφ iφ The differential operator ∇ is given by ∇=

1 ∂ ∂ 1 ∂ ir + iθ + iφ ∂r r ∂θ r sin θ ∂φ

1.5 TENSOR OPERATIONS

9

Some useful differential operations are given by 1 ∂ψ 1 ∂ψ ∂ψ ir + iθ + iφ ∂r r ∂θ r sin θ ∂φ     1 ∂ ∂ ∂ψ 1 ∂ψ 1 ∂ 2ψ ∇ 2ψ = 2 r2 + 2 sin θ + 2 2 r ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂φ 2 1 ∂uφ ∂ 1 ∂ 1 (uθ sin θ ) + ∇ · u = 2 (r 2 ur ) + r ∂r r sin θ ∂θ r sin θ ∂φ   ir r iθ r sin θ iφ 1 ∂/∂r ∂/∂θ ∂/∂φ  ∇ ×u= 2 r sin θ ur ruθ r sin θ uφ

 ∂uθ ∂ 1 (sin θ uφ ) − ir = r sin θ ∂θ ∂φ 

1 ∂ 1 ∂ur + − (ruφ ) iθ r sin θ ∂φ ∂r

 1 ∂ ∂ur + (ruθ ) − iφ r ∂r ∂θ ∇ψ =

In any orthogonal curvilinear system of coordinates, the cross product of a gradient, i.e., ∇ψ, is zero. In other words ∇ × ∇ψ = 0 Also, for a vector, u, one can write ∇ · (∇ × u) = 0 1.5 TENSOR OPERATIONS A second order tensor is a quantity that has nine components that are associated with three orthogonal directions and normal planes. The components of a tensor quantity = T is given by   T11 T12 T13 = T = T21 T22 T23  T31 T32 T33 =

For T to be a stress tensor, each term Tij represents the stress on the i th plane in the j -direction. In general, Tij = Tj i unless the second order tensor is symmetric, where Tij = Tj i . A second order tensor can also be given as =

T =

3
3
i=1 j =1

ii ij Tij

10

MATHEMATICAL PRELIMINARIES

and =

T = i1 i1 T11 + i1 i2 T12 + i1 i3 T13 + i2 i1 T21 + i2 i2 T22 + i2 i3 T23 + i3 i1 T31 + i3 i2 T32 + i3 i3 T33 where i1 , i2 , and i3 are unit vectors. Here, ii ij is called the unit dyad, which follows certain rules when associated with a vector operation such as: ii ij · ik = ii (ij · ik ) = ii δj k ii · ij ik = (ii · ij )ik = ik δij where δij is the Kronecker delta defined as δij = 1

δij = 0

if

i=j

if

i = j

When the components of a second order tensor are formed from components of two vectors u and v, the resulting product is called a dyadic product of u and v, given by uv, where uv =

3 3



ii ij ui vj

i=1 j =1

A unit tensor is defined as =

I=



i

ii ij δij

j

or 1 I = 0 0 =

0 1 0



The magnitude of a tensor is given by

 0 0 1



1 |T | = T2 2 i j ij =

=

=

The addition of tensors T and A or dyadic products simply follows =

=

T +A=



(Tij + Aij ) i

j

1.6 VECTOR AND TENSOR INTEGRAL THEOREMS

11

The multiplication of a tensor or a dyadic product by a scalar gives =

sT =



i

ii ij (sTij )

j

The vector product (or dot product) of a tensor with a vector is given by   

= (T · u) = ii Tij uj   i

j

The vector product (or dot product) of a vector with a tensor is given by    

= (u · T ) = uj Tj i ii   j

i

=

=

=

In general T · u = u · T unless the tensor T is symmetric where Tij = Tj i . In expanded = form one can write for T · u in Cartesian coordinates =

T · u = (Txx ux + Txy uy + Txz uz )ix + (Tyx ux + Tyy uy + Tyz uz )iy + (Tzx ux + Tzy uy + Tzz uz )iz In other orthogonal coordinates, one can simply replace (x, y, z) by the respective new coordinates e.g., with (r, θ, φ) for a spherical coordinate system.

1.6 VECTOR AND TENSOR INTEGRAL THEOREMS 1.6.1 The Divergence and Gradient Theorems If a volume V is enclosed by a surface S, then   (∇ · u)dV = (u · n)dS S

V

where n is the outwardly directed unit normal vector. As u · n = n · u, one can write   (∇ · u)dV = (n · u)dS S

V

=

Two related theorems for scalars, ψ, and tensors, T , can be written as   ∇ψdV = nψdS V

S

12

MATHEMATICAL PRELIMINARIES

and 

=

=

V

(∇ · T )dV =



=

S

(n · T )dS

Clearly, the tensor T can be replaced by a dyadic product. 1.6.2 The Stokes Theorem If a surface S is bounded by a closed curve C, then 

S

n · (∇ × u)dS =



C

(t · u)dC

where t is a unit tangential vector in the direction of integration along path C, n is the unit normal vector to the surface S in the direction that a right-hand screw would move if its head were twisted in the direction of integration along contour C (Bird et al. 2002).

1.7 REFERENCES Bird, R. B., Stewart, W. E., and Lightfoot, E. N., Transport Phenomena, John Wiley, New York, (2002). Hiemenz, P. C., Principles of Colloid and Surface Chemistry, 2nd ed., Marcel Dekker, New York, (1986). Jeffrey, A., Handbook of Mathematical Formulas and Integrals, Academic Press, San Diego, (1995). Lide, D. R., Editor in Chief, Handbook of Chemistry and Physics, 82nd ed., CRC Press, Cleveland, (2001). Probstein, R. F., Physicochemical Hydrodynamics, An Introduction, 2nd ed., Wiley Interscience, New York, (2003). Russel, W. B., Saville, D. A., and Schowalter, W. R., Colloidal Dispersions, Cambridge University Press, Cambridge, (1989).

CHAPTER 2

COLLOIDAL SYSTEMS

The subject of electrokinetic transport is intricately related to the field of colloid and interface science, and hence, a brief primer of colloidal systems and colloidal phenomena is a prerequisite for a sound understanding of electrokinetic processes. 2.1 THE COLLOIDAL STATE The term colloid originates from the Greek word ‘κoλλα’ – meaning glue. In the 19th century, Thomas Graham (1805–1869) coined the terms colloid and crystalloid to classify two types of matter. While colloidal particles form a dispersion or suspension, crystalloids form a homogeneous solution when dissolved in a solvent. Colloidal dispersions are distinct from true (homogeneous) solutions in several ways. In a true solution, the solute is supposed to have lost its identity (consider dissolution of a salt in water: the salt dissociates into its constituent ions, and apparently undergoes a change in property). Colloidal particles, however, retain their identity in a suspension. Therefore, a colloidal suspension is considered a heterogeneous system. The above classification between colloidal dispersions and homogeneous solutions may appear to be ludicrous to a modern physical chemist. We now consider the particulate nature of matter as a universal truth. In this scenario, the classification of colloids from crystalloids (solutes or ions) has become inconsequential, or even redundant. Hence, we generally resort to the simple size based demarcation between colloidal particles and smaller ionic or molecular solutes. Electrokinetic and Colloid Transport Phenomena, by Jacob H. Masliyah and Subir Bhattacharjee Copyright © 2006 John Wiley & Sons, Inc.

13

14

COLLOIDAL SYSTEMS

Figure 2.1. Spherical polystyrene latex particles observed using an atomic force microscope. The individual particles have a diameter of ca. 140 nm.

A colloidal dispersion is loosely defined as a multi-phase system, in which a discrete phase (the dispersed phase) is suspended in a continuous medium called the dispersant. The use of the term “discrete” is crucial in the above definition, since this word imposes a restriction on the size of the dispersed phase relative to the molecules of the dispersant. For instance, if we consider an aqueous dispersion of 25 nanometer (1 nm = 10−9 m) diameter silica particles, the silica particles will be nearly 100 times larger than the water molecules (which have an approximate diameter of 0.276 nm). In this case, although the water molecules are discrete themselves, they are so much smaller than the silica particle, that they will appear as part of a continuous medium relative to the silica particle. In contrast, if we add a salt (say NaCl) to the dispersion, the sodium and chloride ions, with hydrated diameters approximately 0.4–0.5 nm, will be of the same size range as the solvent. In this case, the ions will also appear to be part of a continuum relative to the silica particles. Thus, in order to have a colloidal system, the suspended or dispersed medium should have a size that is approximately one order of magnitude larger than the solvent molecules. In keeping with the scope of the above definition, colloidal particles are usually defined as entities having a size range of 1 nm to about 10 micro-meter (1 µm = 10−6 m). Figure 2.1 depicts spherical polystyrene latex particles of about 140 nm in diameter. Some examples of colloidal systems are outlined in Table 2.1, and typical particle size for some colloidal systems are given in Figure 2.2.

TABLE 2.1. Some Typical Colloidal Systems. Examples

Class

Phase Dispersed

Disperse Systems Fog, mist, smoke, aerosol sprays Industrial smokes Milk, butter, mayonnaise, asphalt, pharmaceutical creams Inorganic colloids (gold, silver iodide, sulphur, metallic hydroxides, etc.), paintsb Clay slurries, toothpaste, mud, polymer lattices Opal, pearl, stained glass, pigmented plastics Froths, foams Meerschaum, expanded plastics Microporous oxides, silica gel, porous glass, microporous carbons, zeolites Macromolecular Colloids Jellies, glue Association Colloids Soap/water, detergent/water, dye solutions Biocolloids Blood Bone Muscle, cell membrane

Liquid aerosol or aerosol Liquid of liquid particlesa Solid aerosol or aerosol Solid of solid particlesa Emulsions Liquid

Continuous Gas Gas Liquid

Sols or colloidal suspensions

Solid

Liquid

When very concentrated called a paste Solid suspension or dispersion Foamc Solid foam

Solid

Liquid

Solid

Solid

Gas Gas

Liquid Solid

Xerogelsd

Gels

Macromolecules Solvent

—

Micelles

Solvent

Corpuscles Serum Hydroxyapatite Collagen Protein structures, thin lecithin films, etc. Coexisting Phases

Three Phase Colloidal Systems (Multiple Colloids) Oil bearing rock Capillary condensed vapors Frost heaving Mineral flotation

Porous rock Porous solid Porous rock or soil Mineral

Oil Liquid Ice Water

Double emulsions

Oil

Aqueous phase

Water Vapor Water Air bubbles or oil droplets Water

Source: Adapted from Everett (1988). a Preferred nomenclature according to IUPAC recommendations. b Many modern paints are more complex, containing both dispersed pigment and emulsion droplets. c In a foam, it is usually the thickness of the film of dispersion medium which is of colloidal dimensions, although the dispersed phase may also be finely divided. d In some cases both phases are continuous, forming interpenetrating networks both of which have colloidal dimensions.

16

COLLOIDAL SYSTEMS

Figure 2.2. Typical particle size ranges in the colloidal domain.

2.2 COLLOIDAL PHENOMENA Colloidal phenomena are concerned with small particles or systems where the ratio of surface area to volume is very large. For spherical particles, the surface area to volume ratio varies as 6/d, where d is the particle diameter. Naturally, as the particle becomes smaller, the surface area per unit volume of the particle increases. Thus, in colloidal systems, in addition to the standard body forces (forces that act over the entire volume of a body) encountered in macroscopic objects, surface forces become important. The surface forces are typically engendered by the interactions occurring at the interfaces between the dispersed phase and the dispersing medium. These surface forces are often dominant in colloidal systems, leading to unique behaviors of colloidal dispersions, which are collectively termed as colloidal phenomena. By and large, the factors which contribute most to the overall nature of a colloidal system are: • • • • •

Particle size and shape Surface properties, both chemical and physical Continuous phase chemical and physical properties Particle–particle interactions Particle–continuous phase interactions

The various forces that enter into the interactions are: • Electric repulsive or attractive force • Attractive or repulsive London–van der Waals force

2.2 COLLOIDAL PHENOMENA

• • • • • •

17

Brownian force Viscous force Inertial force Gravitational force Steric (size-exclusion) forces Surface tension

The electric force between two particles can be derived from Coulomb’s law and it is of the order ǫψs2 , where ǫ is the dielectric permittivity of the continuous phase and ψs is the surface electric potential. The London–van der Waals forces on the atomic scale yield the force between macroscopic bodies known as the dispersion force which is of the order A/a. Here A is the Hamaker constant, which is a function of the properties of both the dispersed and continuous phases. The characteristic length of the particle is given by a. The thermal energy of the molecules manifests itself as a Brownian force of the order kB T /a, where kB is the Boltzmann constant and T is the absolute temperature. The fluid (continuous phase) viscosity gives rise to a viscous force of the order µU a where µ is the viscosity of the continuous phase and U is the particle velocity through the continuous phase. Due to the movement of the bulk particles, an inertial force comes into play which is of the order ρa 2 U 2 , where ρ is the continuous medium density. The gravitational body force on a particle is of the order a 3gρ where g is the acceleration due to gravity and ρ is the density difference between the particle and the continuous phase. Surface tension arises from the interaction between the two phases represented by the particle and the continuous medium. This force is of the order γ a, where γ is the surface (interfacial) tension of the particle in the medium. Table 2.2 gives the order of magnitude values for the relative significance of the force cited above (Russel et al., 1989). For the particular values chosen, it is clear that the ratio of the inertial to viscous forces is not important and that the ratio of repulsive electric force to Brownian force is fairly high. This would indicate a stable colloidal system. However, it should be recognized that the electrical and attractive forces between particles are greatly influenced by the separation distance between the particles. Consequently, the analysis cannot be considered as being complete. Table 2.2 shows that the ratio of various forces contains the particle dimension, which can vary from 10−8 to 10−5 m, and consequently, can affect the nature of the interactions. As the surface to volume ratio is of the order 1/a, it becomes clear that for very small particles, the surface to volume ratio can be significantly large and a high percentage of the molecules in such a particle will lie within or close to the region of inhomogeneity associated with the particle/medium interface. These molecules have properties different from those in the bulk phases more distant from the interface. It is no longer possible to describe a colloidal system (dispersed and continuous phases) simply in terms of the sum of contributions from molecules in the bulk phases, calculated as if both phases had the same properties as they would have

18

COLLOIDAL SYSTEMS

TABLE 2.2. Magnitudes of the Characteristic Force: T = 300 K, a = 1 µm, µ = 10−3 Pa s, U = 1 × 10−6 m/s, ρ = 103 kg/m3 , ρ/ρ = 10−2 , g = 10 m/s2 , A = 10−20 J, ψs = 0.05 V, ǫ = 8.85 × 10−10 C/Vm, γ = 0.1 N/m, kB = 1.381 × 10−23 J/K. ǫψs2 kB T /a A/a kB T /a kB T /a µU a a 3 gρ µU a ρa 2 U 2 µU a γa a 3 gρ

electrical force Brownian force attractive force Brownian force Brownian force viscous force gravitational force viscous force inertial force viscous force surface tension gravitational force

≈ 103 ≈1 ≈1 ≈ 10−1 ≈ 10−6 ≈ 109

Source: Adapted from Russel et al. (1989).

in the individual bulk state. A significant and dominating contribution comes from the molecules residing at the interface. This is why surface chemistry plays an important role in colloid science and why colloidal properties become important as the particle decreases in size, say below 10 µm (Everett, 1988). The following example demonstrates how the ratio of the number of the “surface molecules” to the total molecules increases with the decreasing ratio of particle size to the molecule size. Consider a particle in the shape of a cube having a side length of L. Assume that the particle material is made up of molecules having the shape of a cube whose side length is S. The total number of molecules Nt contained in the particles is given by
3 L Nt = S

(2.1)

The number of surface particles is given by Ns = 2


 2


2 L L L L −2 −2 +2 +2 S S S S

for L/S ≥ 2

(2.2)

Figure 2.3 shows the subdivision of the particle. The ratio of the number of surface molecules to the total number of molecules is then given by

3
2 S S S Ns =6 +8 − 12 Nt L L L

(2.3)

2.2 COLLOIDAL PHENOMENA

19

Figure 2.3. Subdivision of a cube by its molecules.

For the case of S/L ≪ 1, Eq. (2.3) gives
 S Ns =6 Nt L

(2.4)

For L = 10 nm and S = 0.3 nm, Eq. (2.3) gives Ns /Nt = 0.17. This indicates that 17% of the molecules reside on the particle surface, i.e., on the interface of the particle and the medium. Such a high percentage would indicate that the properties of the molecules are important in describing the behavior of a colloidal particle. Figure 2.4 gives the variation of Ns /Nt with S/L.

Figure 2.4. Ratio of surface to total molecules as a function of size ratio.

20

COLLOIDAL SYSTEMS

Figure 2.5. A particle straddling an air–water interface.

Another example is that of a particle straddling an interface. It illustrates the significance of the particle size. Consider a cube of side length L and density ρp straddling the interface of air and water. The water density is ρw and the air density is ρa . Figure 2.5 shows such a particle. Assuming a downward force is positive, the forces acting on the cube are given by Surface Tension:

−4Lγ cos θ

Gravitation: L3 ρp g Buoyancy:

−DL2 ρw g − (L − D)L2 ρa g

At equilibrium, the sum of all the forces is zero, hence L3 ρp g − 4Lγ cos θ − DL2 ρw g − (L − D)L2 ρa g = 0

(2.5)

where D is the submerged length and θ is the static contact angle which is related to the surface tensions of the particle-air, particle-water, and water-air. As ρa /ρw ≪ 1, Eq. (2.5) leads to ρp 4γ cos θ D = (2.6) − 2 L ρw L ρw g and ρp D 4 cos θ = − (2.7) L ρw Bo where ρw gL2 gravitational force = = Bo γ surface tension force

21

2.3 STABILIZATION OF COLLOIDS

Figure 2.6. A floating particle at an air–water interface having a small Bond number.

The dimensionless group Bo is known as the Bond number. For given physical properties ρp , ρw , γ , and cos θ, and constant g, the Bond number → 0 as the particle dimension L → 0. For 0 < D/L < 1, the particle floats on the surface. The (D/L) value is a balance between (ρp /ρw ) and 4 cos θ/Bo. For a large particle, Bo is large and Eq. (2.7) leads to D/L = ρp /ρw > 1. This is the situation where the particle sinks into the water. This criterion is only related to the density ratio. For the case of a very small particle, i.e., L → 0, and Bo → 0, Eq. (2.7) gives D 4 cos θ ≈− L Bo

(2.8)

A negative (D/L) value signifies that a small particle will always float on the water surface irrespective of the density ratio. This is shown in Figure 2.6. Consequently, it is clear that the particle size has a major role in determining its behavior.

2.3 STABILIZATION OF COLLOIDS Colloidal particles can be stabilized against coagulation (or flocculation) by electrostatic repulsion due to the presence of ions near their surfaces or by steric effects arising from polymer chains being attached to the surface of particles. The properties of a colloidal system, i.e., its rheology, shear stability, and stability to added electrolyte or polymers are much affected by the nature of the stabilizing mechanism (Walbridge, 1987). Colloidal particles can be stabilized by the electrostatic forces that arise as a result of the charged particle surface and the presence of an associated diffuse atmosphere (double layer) of counterions. Surface charge can arise from different causes, e.g., surface ionization and physically adsorbed ionized surfactant groups. Electrostatic stabilization plays a dominant role in aqueous systems. Electrostatically stabilized particles can be flocculated by the addition of an electrolyte and by shear. Figure 2.7 shows the effect of electrolyte addition on the total interaction potential energy φ of a colloidal particle. A positive value of the derivative of the interaction potential energy with respect to the separation distance indicates an attractive force. Figure 2.7 shows that increasing the electrolyte concentration changes the shape of the interaction potential energy leading to weak repulsive force and a possible attractive force at a large separation distance between the particles.

22

COLLOIDAL SYSTEMS

Figure 2.7. Variation of the total potential at different electrolyte concentrations for charge stabilized particles.

A colloidal dispersion can be stabilized in either an aqueous or non-aqueous continuous phase by solvated polymeric moieties adsorbed on the colloidal particle surface. The polymeric chains attached to the surface can be regarded as a barrier around each particle, preventing their close approach to each other. For sterically stabilized colloidal systems, the continuous phase (medium) must be a good solvent to the attached polymer with nearly complete surface coverage as shown in Figure 2.8(a). The case of a poor solvent is depicted in Figure 2.8(b). The total potential energy for sterically stabilized particles is shown in Figure 2.9. Sterically stabilized colloidal particles are usually very stable over a wide range of particle sizes and shear rates, and at high dispersed phase concentrations. They can

Figure 2.8. (a) Highly sterically stabilized colloidal particle (polymer is in a good solvent); (b) poorly sterically stabilized colloidal particles (polymer is in poor solvent).

2.4 PREPARATION OF COLLOIDAL SYSTEMS

23

Figure 2.9. Variation of the total interaction potential energy for sterically stabilized particles.

be flocculated by changing the solvency of the continuous phase or by desorbing the attached polymeric moieties. In practice, molecular weights above 1000 are desirable (Walbridge, 1987).

2.4 PREPARATION OF COLLOIDAL SYSTEMS There are two fundamentally different ways in which colloidal dispersions can be made: either by breaking down (splitting) bulk coarse matter to colloidal dimensions or by building up molecular aggregates to colloidal size. The first method is referred to as the dispersion method and the second is referred to as the condensation or nucleation method. 2.4.1

Dispersion Methods

Energy changes are associated with the breakup of a coarse particle. The free energy change associated with creation of new surface area, A, is given by G = γ A

(2.9)

where γ is the solid material surface tension and G is the change in Gibbs free energy due to the creation of new surface area. G is then the work required to create the new surface area and it is directly related to γ and A (Everett, 1988). The diameter of particles obtained in the grinding or milling of a solid, either in a mortar or in a ball mill, is about 1 to 5 µm. The size of the colloidal particles can be reduced by grinding the solids in a liquid having a low surface tension. A further decrease in size can be achieved by the addition of a surface-active electrolyte

24

COLLOIDAL SYSTEMS

to the grinding medium. These electrolytes, such as soaps or long chain alkylarylsulphonates, are adsorbed on the particle surfaces and hence tend to stabilize the dispersion of the colloidal particles. Particle sizes of the order of 0.1 µm can be achieved by wet milling (Jirgensons and Straumanis, 1962). Formation of emulsions by the “breaking down” of one liquid in the presence of another can be achieved either by simple stirring or by applying high shear to the two liquid phases. Colloidal mills or high speed homogenizers are commonly used to prepare a liquid-in-liquid dispersion, i.e., an emulsion. The success of an emulsification process depends, to a large extent, on the interfacial tension between the two liquid phases and on the stabilizing force present. Emulsifying agents, i.e., agents that lower interfacial tension, are usually required for successful emulsification. In addition to the mechanical approach used in breaking up solids and liquids, ultrasonic waves can be used for the breaking up of solids in the emulsification process. The velocity of sound v is related to the wavelength λ and the frequency f by v = fλ (2.10) Ultrasonic vibrations can be easily generated in the range of 25 kHz–2 MHz with commercially available equipment. An oscillating quartz disk is able to produce very high intensity ultrasonic vibrations of up to 105 W/m2 . This intensity is about 1010 times larger than the intensity produced by a loud radio. Such an energy level can disintegrate coarse particles and form emulsions (Jirgensons and Straumanis, 1962). It is interesting to note that with proper particle size and vibration frequency, the reverse process of solids agglomeration or de-emulsification can occur. 2.4.2

Condensation Methods

Colloidal particles can be made by combining small molecules into larger units. This can be achieved either by a decrease in solubility or through chemical methods. The chart in Figure 2.10 illustrates the types of condensation methods used in making of a colloidal system (or sol). A simple way to prepare a colloidal system is to pour a true molecular solution of a solute into another liquid in which the solute is practically insoluble. A sulphur sol can be prepared in this way by decreasing the sulphur solubility due to a change in the solvent. Sulphur is dissolved in alcohol and then the solution is diluted with

Figure 2.10. Types of condensation methods used for preparation of colloidal systems (sols).

2.4 PREPARATION OF COLLOIDAL SYSTEMS

25

water in which the sulphur is much less soluble. Various hydrosols of resins and fats are prepared similarly. Instead of reducing the solubility by changing the solvent, one can decrease the temperature. A sol of ice in pentane can be prepared by cooling pentane that contains traces of water. Chemical methods provide excellent means for preparing colloidal systems. The basic idea is to perform a chemical reaction in which an insoluble or practically insoluble substance is formed so that the solid remains dispersed as small particles. For example, sols of hydroxides can be prepared by hydrolysis. Specifically, colloidal ferric hydroxide is prepared by the hydrolysis of ferric chloride at the boiling point: FeCl3 + 3H2 O → Fe(OH)3 + 3HCl Methods of preparation and characterization of monodisperse metal hydrous oxide sols, i.e., colloidal dispersions, are given by Matijevic (1976). Monodisperse silica spheres can be prepared using techniques advanced by Stöber et al. (1968), van Blaaderen and Vrij (1993), and Nyffenegger et al. (1993). Some recipes for preparation of some simple sols are given below (Everett, 1988). Gold Sol: Add 1 cm3 of a 1% solution of gold chloride (HAuCl4 ·3H2 O) to 100 cm3 of distilled water, bring to boil and add 2.5 cm3 of 1% sodium citrate solution. Keep the solution just boiling. After a few minutes observe the appearance of a blue coloration, followed by the formation of a ruby-red gold sol. Sulphur Sol: Rapidly mix 4 mM (millimolar, 1 mM = 0.001 mol dm−3 ) sodium thiosulphate solution and 4 mM hydrochloric acid. The mixture becomes cloudy after a few minutes and then develops to an opaque white dispersion of colloidal sulphur. Silver Bromide Sol: Mix equal amounts of 20 mM sodium bromide solution to 18 mM silver nitrate solution. A colloidal dispersion of silver bromide is formed immediately. A silver iodide sol may be prepared in a similar manner. Ferric Hydroxide Sol: Add 2 cm3 of a 30% solution of ferric chloride slowly, with stirring, to 500 cm3 of boiling distilled water. A clear reddish-brown dispersion of ferric hydroxide is formed. Polymeric Dispersions: Polymeric dispersions can be prepared by emulsion polymerization, dispersion polymerization, and suspension polymerization (Walbridge, 1987). In emulsion polymerization, a monomer is emulsified in a non-solvent (usually water) in the presence of a surfactant. A water-soluble initiator is added. Particles of the polymer form and grow in the aqueous continuous phase. Production of the particles proceeds until the monomer is used up. In dispersion polymerization, monomer, initiator, stabilizer, and solvent initially form a homogeneous solution. Polymer precipitates when the solubility limit is exceeded. Polymer particles continue to grow until the monomer is used up.

26

COLLOIDAL SYSTEMS

In suspension polymerization, the monomer is emulsified in the medium using a surfactant. The initiator is often dissolved within the monomer droplet. The droplet is gradually converted into the insoluble particles, but no new particles are formed. The size distribution of the dispersion becomes that of the original monomer droplets. In most of the processes involving preparation of a colloidal system, the formation of the sols passes through nucleation and growth stages. Control of such stages would determine the size of the colloidal particles and their size distribution.

2.5 PURIFICATION OF SOLS In the previous section, we dealt with methods of making colloidal systems of solids or of immiscible liquids in either an aqueous or non-aqueous continuous phase. Prepared sols of solid particles that are insoluble in water or in a solvent usually contain some contamination having low molecular weight. These contaminants can be removed by various methods such as dialysis, ultrafiltration, and electrodialysis. Dialysis, in its simplest form, involves placing the sol in a container having one end covered with a semi-permeable membrane. The membrane side of the container is placed in water (or solvent) as shown in Figure 2.11. The membrane is permeable to the solvent and the other small molecular weight impurities, but impermeable to the colloidal particles. Diffusion of the impurities through the membrane leads to the eventual depletion of the impurities within the prepared sols (Overbeek, 1952). Ultrafiltration is a pressure driven membrane separation process. The solvent and low molecular weight impurities pass through the membrane but the membrane rejects particulate matter. Ultrafiltration is similar to common filtration processes except that the membrane pores are very small. The applied pressure is usually in the range of 200– 1000 kPa. Ultrafiltration is not a method of purification but rather of concentration. Upon the rejection of the solvent, together with the associated impurities, fresh solvent is added to the sol and ultrafiltration is once again applied to the sol. The addition of the fresh solvent acts as a washing medium and makes it possible to obtain a sol with little impurities.

Figure 2.11. A simple dialysis cell.

2.6 A HISTORICAL SUMMARY

27

Figure 2.12. Electrodialysis cell pair. Sol purification from Na+ and Cl− ions. Adapted from Probstein (2003).

Electrodialysis is a membrane process in which dissolved ions are removed from an aqueous solution through membranes under the driving force of a dc electric field. Electrodialysis membranes are ion-exchange membranes (Probstein, 2003). An electrodialysis cell has alternating anion and cation exchange membranes. The sol to be purified is made to flow between the membranes. An anion exchange membrane allows the passage of the anions but prevents passage of the cations. Conversely, a cation exchange membrane allows the cations to pass, while retaining the anions. In this manner, it is possible to concentrate the undesirable electrolyte and to deplete the sol from the electrolyte (e.g., NaCl). This is shown in Figure 2.12. The chamber containing the electrolyte-free sol is called the dialysate channel and the chamber containing the electrolyte-rich sol is called the concentrate channel.

2.6 A HISTORICAL SUMMARY As stated in the beginning of this chapter, Thomas Graham classified colloids and crystalloids as two types of matter. His classification was based on diffusion experiments through membranes. Colloidal systems were observed and studied for at least two centuries prior to Graham’s studies, since about the seventeenth century, when alchemists produced sols by treating gold chloride solutions with reducing agents. Observation of the behavior of colloidal particles in a suspension dates back to the works of Robert Brown (1773–1858). He studied the erratic motion of particles in

28

COLLOIDAL SYSTEMS

pollen grains suspended in water under a microscope in 1827. A detailed description of his experiments and a recreation of the original experiments is given by Ford (1992). The molecular basis of Brownian motion was settled by Perrin (1870– 1942) in his book “Brownian Motion and Molecular Reality” (1910). The theory of Brownian motion was presented by Einstein (1879–1955). His papers on Brownian movement, published between 1905 and 1911, have been translated into English and the compilation is available as a book edited with notes by Fürth (1926). Shortly after Einstein’s first work on Brownian motion, Langevin (1872–1946) formulated a theory in which the minute fluctuations in the position of a particle were due explicitly to a random force (1908). Langevin’s approach proved to have great utility in describing molecular fluctuations in colloidal systems, including non-equilibrium thermodynamics. The early experimental observations on dynamics of colloidal systems were intricately related to the development of optical microscopy. Following Faraday’s discovery that small particles could be detected by focussing light rays into a cone, Zsigmondy and Siedentopf invented the ultramicroscope in 1903. Studies with this instrument probed the nature of the erratic Brownian motion of individual particles. The discovery that naturally occurring colloidal particles were charged was made by Reuss in 1809, who observed the motion of clay particles in an electric field. Schulz (1882) and Hardy (1900) elucidated the role of added electrolytes in suppressing the effects of charge and promoting coagulation. Their work provided strong evidence that stability of aqueous dispersions derived from electrostatic repulsion. Smoluchowski (1872–1917) provided the celebrated formula relating the surface electric potential [more appropriately, the zeta (ζ ) potential] to the electrophoretic mobility (1903). A significant amount of the early theoretical developments in electrokinetic transport phenomena can be attributed to Smoluchowski and Helmholtz (1821–1894). Gouy (1910) and Chapman (1913) presented the theory for the screening of surface charge by the diffuse layer of counterions, thereby relating the thickness of the diffuse layer to the ionic strength of the solution. Smoluchowski (1917) deduced expressions for the rate of formation of small aggregates by Brownian and shear-induced collisions between particles, which related the flocculation rates to the screening of the electrostatic repulsion with excess electrolyte. Although the theoretical interpretation of colloid stabilization employing electrostatic repulsion was becoming more formal in the early twentieth century, the nature of the attractive force that causes destabilization of colloids and subsequent flocculation was not explored formally until the works of de Boer (1936) and Hamaker (1937). They developed a theory to obtain the attractive van der Waals interactions between colloidal particles by pairwise summation of the fundamental atomistic (Lennard– Jones) interaction potentials. This led to the representation of the total interparticle interaction potential as the sum of the attractive van der Waals and the repulsive electrostatic interactions. The first formal theory of colloidal stability was presented independently by Derjaguin and Landau (1941) in the Soviet Union, and Verwey and Overbeek (1948) in the Netherlands, who used the total interparticle interaction as a linear combination of the van der Waals and electrostatic interactions between a pair of particle. The DLVO theory, named after these four scientists, forms a basis of modern

2.7 ELECTROKINETIC PHENOMENA IN MODERN COLLOID SCIENCE

29

theoretical colloid science, and remains a cornerstone for theoretical elucidation of colloid transport phenomena.

2.7 ELECTROKINETIC PHENOMENA IN MODERN COLLOID SCIENCE Electrokinetic transport phenomena have been an important aspect of colloid science since the early twentieth century. Despite the fact that the first observations of electrokinetic phenomena date back to the studies conducted by Reuss in 1809, the first theoretical developments of electrokinetic transport are attributed to Helmholtz (1879) and Smoluchowski (1903). The development of the subject following these early works has been summarized by Dukhin and Derjaguin (1974). Considerable attention has since been conferred on four key electrokinetic phenomena, namely, streaming potential, electroosmotic flow, electrophoresis, and sedimentation of charged suspensions. Several books provide details of these developments (Russel et al., 1989; Probstein, 2003), although attention has mostly been devoted to dilute suspensions and single charged particles. The last two decades of the past century have seen a steady improvement in the understanding of these phenomena in concentrated and confined systems. In this book, we will discuss these developments in considerable detail. Electrokinetic processes involving colloidal systems have received heightened attention over the past decade, particularly due to the rapid developments associated with biological separations, micro-electromechanical systems (MEMS), and nano-scale separation processes. The renewed interest in electrokinetic phenomena primarily stems from the requirement of moving fluids in confinements (or channels) that are extremely narrow. Clearly, fluid handling devices, particularly pumps, operating in the macroscopic regime are incapable of handling the requirements of the microscopic transport realm, the single biggest challenge being the enormous pressure gradients necessary to drive a fluid through a capillary of microscopic radius. Consequently, there has been an immense effort toward exploration of alternative means of fluid transport in microscopic channels. Electrokinetic transport is perhaps the most suitable candidate for providing the motive force to the fluids in microscopic domains without requiring enormous pressure gradients. Electrophoresis is one of the major processes utilized in biological separations. Biocolloids can be manipulated in a small environment by applying an electric field. This principle is utilized to separate proteins of different sizes or charges, DNA, and cells. Electrokinetic phenomena like electrophoresis and streaming potential are routinely employed to estimate the charge of colloidal particles in different media. Electrical forces are used to cut, transport, mix, separate, and manipulate microscopic liquid droplets in microfluidic chips. Processes like electrowetting (EW) or electrowetting on a dielectric (EWOD) are being used more frequently for moving fluids (Cho et al., 2003). There is a tremendous surge in application of alternating current (ac) electrokinetic phenomena like dielectrophoresis as separation methods for cells and colloidal particles (Hughes, 2003). In all these modern applications, we

30

COLLOIDAL SYSTEMS

need to understand colloidal and electrokinetic transport phenomena in a new light. Unlike the historical perspective of colloid science, which dealt with these phenomena to understand colloidal behavior in bulk systems, modern colloid science has to deal with overall system sizes that are often comparable to the colloidal particle dimensions envisioned by the early colloid scientists. This necessitates a re-evaluation of many traditional theories and concepts of colloid science and electrokinetics. This book aims to bridge the gap between the traditional understanding of electrokinetic transport phenomena in colloidal systems and the modern approaches evolving from the necessity of interpreting electrokinetic and colloid transport behavior in extremely small domains. It provides a detailed evaluation of the fundamental concepts underlying electrokinetic transport processes, while pointing out the modifications in traditional theories needed to address the modern challenges posed by micro-scale transport devices. Accordingly, the book builds up on the basic construct of electrokinetic transport processes in Chapters 3 to 6 by providing a detailed background on electrostatics in vacuum and pure dielectrics, electrical double layers in the presence of free charge, and basic transport equations for charged electrolyte solutions and colloidal suspensions in presence of an external electric field. Following a brief overview of the four key electrokinetic phenomena in Chapter 7, detailed analyses of these four major electrokinetic phenomena, namely, electroosmosis, streaming potential, electrophoresis, and sedimentation potential, are given in Chapters 8 to 10. Following this, we introduce the van der Waals forces and the DLVO theory of colloid stability in Chapter 11. Chapters 12 and 13 present the two important aspects of colloid transport phenomena, namely, coagulation and deposition, respectively. In Chapter 14, detailed numerical approaches for modeling electrokinetic transport phenomena in micro-scale processes are described. In the final chapter of the book, we present some practical applications of the theoretical principles outlined and developed in the book, including membrane filtration processes, environmental cleanup, extraction of bitumen, and microfluidic applications.

2.8 NOMENCLATURE a Bo D d f g G kB L Ns Nt S T

particle radius, m Bond number, dimensionless immersed depth in water, m diameter of sphere, m frequency of sound, s−1 acceleration due to gravity, m/s2 free energy, J Boltzmann constant, J/K cube side length, m total number of surface molecules total number of inner molecules molecule side length, m absolute temperature, K

2.9 REFERENCES

U v

31

particle velocity, m/s speed of sound, m/s

Greek Symbols γ ǫ φ λ µ θ ρ ψs

interfacial tension, N/m dielectric permittivity of medium, C/Vm total interaction potential energy, J sound wave length, m fluid viscosity, Pa.s angle related to contact angle = π − contact angle density, kg/m3 surface electric potential, V

2.9 REFERENCES Chapman, D. L., A contribution to the theory of electroencapillarity, Phil. Mag., 25, 475–481, (1913). Cho, S. K., Moon, H. J., and Kim, C. J., Creating, transporting, cutting, and merging liquid droplets by electrowetting-based actuation for digital microfluidic circuits, J. Microelectromech. Systems, 12, 70–80, (2003). de Boer, J. H., The influence of van der Waals forces and primary bonds on binding energy, strength and orientation, with special reference to some artificial resins, Trans. Faraday Soc., 32, 10–38, (1936). Derjaguin, B. V., and Landau, L. D., Theory of the stability of strongly charged lyophobic colloids and the adhesion of strongly charged particles in solutions of electrolytes, Acta Physicochim. URSS, 14, 633–662, (1941). Dukhin, S. S., and Derjaguin, B. V., Electrokinetic Phenomena, in Surface and Colloid Science, vol. 7, E. Matijevic (Ed.), Wiley, (1974). Einstein, A., Investigations on the Theory of the Brownian Movement, R. Fürth (Ed.), Dutton, NY, (1926) (also a Dover publication, 1956). Everett, D. H., Basic Principles of Colloid Science, Royal Society of Chemistry, London, (1988). Ford, B. J., Brownian movement in clarkia pollen: A reprise of the first observations, The Microscope, 40, 235–241, (1992). Gouy, G., Sur la constitution de la electrique a la surface d’un electrolyte, J. Phys. Radium, 9, 457–468, (1910). Hamaker, H. C., London–van der Waals attraction between spherical particles, Physica, 4, 1058–1072, (1937). Hardy, W. B., A preliminary investigation of the conditions which determine the stability of irreversible hydrosols, Proc. Roy. Soc. Lond, 66, 110–125, (1900). Helmholtz, H. V., Studien uber elctrische grenschichten, Ann. der Physik und Chimie, 7, 337–387, (1879). Hughes, M. P., Nanoelectromechanics in Engineering and Biology, CRC Press, Boca Raton, (2003).

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Jirgensons, B., and Straumanis, M. E., A Short Textbook of Colloid Chemistry, McMillan Co., London, (1962). Langevin, P., Theory of Brownian motion, C. R. Acad. Sci., 146, 530–533, (1908). Matijevic, E., Preparation and characterization of monodisperse metal hydrous oxide sols, Prog. Colloid Polymer Sci., 61, 24–25, (1976). Nyffenegger, R., Quellet, C., and Ricka, J., Synthesis of fluorescent, monodisperse, colloidal silica particles, J. Colloid Interface Sci., 159, 150-157, (1993). Overbeek, J. Th. G., Phenomenology of lyophobic systems, Chapter II, in Colloid Science, H. R. Kruyt (Ed.), Elsevier, Amsterdam, (1952). Perrin, J., Brownian Motion and Molecular Reality, Taylor and Francis, London, (1910). Probstein, R. F., Physicochemical Hydrodynamics, An Introduction, 2nd ed., Wiley Interscience, New York, (2003). Russel, W. B., Saville, D. A., and Schowalter, W. R., Colloidal Dispersions, Cambridge University Press, Cambridge, (1989). Schulz, H., Schwefelarsen im wasseriger losung, J. Prakt. Chem., 25, 431–452, (1882). Smoluchowski, M. von, Contribution a la theorie de l’endosmose electrique et de quelques phenomenes correlatifs, Bull. International de l’Academie des Sciences de Cracovie, 8, 182–200, (1903). Smoluchowski, M. von, Versuch einer mathematischen Theorie der Koagulationkinetik kollider losungen, Z. Phys. Chem., 92, 129–168, (1917). Stöber, W., Fink, A., and Bohm, E., Controlled growth of monodisperse silica spheres in micron size range, J. Colloid Interface Sci., 26, 62–69, (1968). van Blaaderen, A., and Vrij, A., Synthesis and characterization of monodisperse colloidal organo-silica spheres, J. Colloid Interface Sci., 156, 1–18, (1993). Verwey, E. J. W., and Overbeek, J. Th. G., Theory of Stability of Lyophobic Colloids, Elsevier, Amsterdam, (1948). Walbridge, D. J., Preparation of liquid-solid dispersions, Chapter 2, in Solid/Liquid Dispersions, Tadros, Th. F. (Ed.), Academic Press, London, (1987).

CHAPTER 3

ELECTROSTATICS

It was pointed out earlier that most surfaces attain a surface charge when immersed in an electrolyte solution. The electrostatic forces arising from the surface charge are essential for stabilizing colloidal suspensions and they play a central role in biological systems and industrial processes. In order to appreciate and better understand the role of charged surfaces immersed in an electrolyte solution where free electric charges are present, we will deal in this chapter with basic electrostatics in free space, i.e., in vacuum, and in materials called dielectrics in the absence of free charge. In other words, we will deal with physical situations where there is no electric current.

3.1 BASIC ELECTROSTATICS IN FREE SPACE 3.1.1

Fundamental Principles of Electrostatics

One can state that there are four fundamental principles (or laws) of electrostatics (Feynman et al. 1964; Slater and Frank, 1969; Eyges, 1980; Griffiths, 1989). The first principle is that of charge conservation. It simply states that the total charge on an isolated body cannot be changed. In other words, for an isolated body,
Qi = constant, where Qi is the i th charge contained in the body. The second law is that charge is “quantized”. This means that the total charge on a body is an integral multiple of a fundamental charge. Here, the magnitude of the charge carried by an electron (1.602 × 10−19 Coulombs) is the fundamental elementary Electrokinetic and Colloid Transport Phenomena, by Jacob H. Masliyah and Subir Bhattacharjee Copyright © 2006 John Wiley & Sons, Inc.

33

34

ELECTROSTATICS

y Q1

R

Q2

x z Figure 3.1. Two point charges separated by distance R in vacuum.

charge, often denoted by the symbol e. Consequently, a charge is given by Ne where N is an integer number and e is the fundamental charge. The third principle is Coulomb’s law that describes the force between two point charges. Consider two stationary point charges of magnitude Q1 and Q2 in free space separated by a distance R as shown in Figure 3.1. Coulomb’s law states that the mutual force between these two charges, F , is given by Q1 Q2 F = (3.1) 4π ǫo R 2 The force is in Newtons (N), the charge is in Coulombs (C), and the separation distance is in meters (m). Here, ǫo is the permittivity of free space, i.e., vacuum, its value in SI units being 8.854 × 10−12 C/Vm or C2 /Nm2 , where V is an abbreviation for Volts, the SI unit for electric potential. The significance of Eq. (3.1) is that the force is proportional to the product of the charges and it is inversely proportional to the square of the separation distance between the charges. As force is a vector, Eq. (3.1) representing the mutual force between the two charges needs more clarification. Accordingly, Coulomb’s law can be formally written in a vectorial form as Q1 Q2 R12 F12 = (3.2) 4π ǫo |R12 |3 Here F12 is the force exerted by charge Q1 on charge Q2 , or analogously, the force felt by charge Q2 due to the charge Q1 . The force acts along a straight line joining the two point charges in the direction of charge Q1 to charge Q2 . This direction is represented by the direction of the vector R12 , which is often termed as the separation (or displacement) vector for charges Q1 to Q2 . The position vectors of the two charges Q1 and Q2 are given as r1 and r2 , respectively. The separation vector R12 is related to the position vectors as R12 = r2 − r1

(3.3)

and its magnitude provides the linear distance between the two charges |R12 | = R

(3.4)

3.1 BASIC ELECTROSTATICS IN FREE SPACE

35

y Q1

R12

r1

Q2

r2 x z Figure 3.2. Position and displacement vectors for two point charges.

The position and separation vectors are shown in Figure 3.2. Denoting the separation vector as R12 = |R12 | i12

(3.5)

where i12 is a unit vector in the direction of the separation vector R12 directed from point charge 1 to point charge 2, Eq. (3.2) can be written as F12 =

Q1 Q2 i12 4π ǫo |R12 |2

(3.6)

Equations (3.2) and (3.6) are identical. By definition, Eq. (3.6) provides that F12 = −F21

(3.7)

or F21 =

Q1 Q2 i21 4π ǫo |R12 |2

(3.8)

where the unit vector i21 is directed from point charge Q2 to point charge Q1 . It should be noted that apart from the direction of the separation vector R12 , the sign of the product Q1 Q2 also dictates the direction of the force. The electrostatic forces for the cases of Q1 Q2 > 0 and Q1 Q2 < 0 are shown in Figures 3.3(a) and 3.3(b), respectively. The case Q1 Q2 > 0 represents charges of the same sign, whereas the case Q1 Q2 < 0 represents charges with opposite signs. For the case of Q1 Q2 > 0 the forces are repulsive, while they are attractive for Q1 Q2 < 0. The fourth principle of electrostatics is that of superposition. It states that the force between any two charges is unaffected by the presence of other charges. In other words, Coulomb’s law between any two point charges is not affected by the presence of any other charges. This law may sound trivial, but it forms the cornerstone of electrostatics. The use of superposition principle is not confined to electrostatics as it is also applicable to Newtonian mechanics.

36

ELECTROSTATICS

Q1Q2 < 0

Q1Q2 > 0 y

y F21

Q1 F21

Q1

Q2

R12

r1

F12

r1

r2

F12

R12

Q2

r2 x

x

z

z (a)

(b)

Figure 3.3. (a) Repulsive forces for Q1 Q2 > 0 case. (b) Attractive forces for Q1 Q2 < 0 case.

3.1.2

Electric Field Strength

Consider many point charges Q1 , Q2 , . . . , Qn at distances R1A , R2A , . . . , RnA , respectively, from a positive test charge that we designate as QA . The geometry is shown in Figure 3.4. Utilizing the superposition principle, the total force (in a vectorial form) exerted by the point charges i = 1, 2, . . . , n on the test point charge QA is given by F = F1A + F2A + · · · + FnA   Q1 QA i1A Q2 QA i2A Qn QA inA 1 + + · · · + = 4π ǫo |R1A |2 |R2A |2 |RnA |2

(3.9)

where iiA is the unit vector along point charges Qi and QA . Here, RiA represents the displacement vector between point charges Qi and QA , such that (3.10)

RiA = rA − ri

where rA and ri represent the position vectors of the point charges QA and Qi , respectively. Equation (3.9) can be rewritten in a concise form as F=

n QA  Qi iiA 4π ǫo i=1 |RiA |2

y

Q2 Q1 r1

R1A rA

(3.11)

Qn QA RiA

ri

Qi x

z Figure 3.4. Point charges in free space.

3.1 BASIC ELECTROSTATICS IN FREE SPACE

37

ir

y A r

x

Q

z Figure 3.5. Electric field strength at position A due to a point charge, Q, located at the origin.

Equation (3.11) leads us to the definition of the electric field strength. Formally, the electric field strength, E, is defined as the force per unit positive test charge. In mathematical notation, F (3.12) E= QA where F is the electric force on a test point charge QA . Following Eq. (3.12), the electric field strength at the location of the test charge QA is obtained by rewriting Eq. (3.11) as E=

n F 1  Qi = iiA QA 4π ǫo i=1 |RiA |2

(3.13)

The electric field strength, E, is due to the point charges Qi ; it acts at the location of the test charge QA and does not depend on the magnitude of the charge QA . It has units of V/m. As a special case of Eq. (3.13), the electric field strength at a location r due to a point charge, Q, located at the origin is given by E=

Q ir 4π ǫo r 2

(3.14)

where ir is a unit vector in the radial (r) direction of the spherical coordinate system. The geometry of a point charge at the origin is shown in Figure 3.5. As the electric field is a vector quantity, one needs to consider its direction as well. As was observed in the case of force, the direction of the electric field is dictated by the direction of the unit separation vector as well as the sign of the charge. To illustrate this, we consider the special case of a point charge Q at the origin, given by Eq. (3.14). When this charge is negative, the field appears to converge to the point charge (the vectors point toward the charge Q), while if Q is positive, the field appears to diverge from the point charge (the vectors point away from the charge Q). One can generalize Eq. (3.13) by replacing the discrete point charges with a continuum bearing infinitesimal charges pertaining to a line, surface, and volume.

38

ELECTROSTATICS

Figure 3.6. Geometry of a line, surface, and volume bearing distributed charges.

The relevant electric field strength at a location A becomes: For a line charge  1 i E= λ dl 4π ǫo line R 2

(3.15)

For a surface charge 1 E= 4π ǫo



i qs dS R2

(3.16)

1 E= 4π ǫo



i ρ dV R2

(3.17)

surface

For a volume charge volume

where the unit vector i is directed from the charge bearing element to point A and R is the distance between them as shown in Figure 3.6. We define: λ is the line charge density, C/m. qs is the surface charge density, C/m2 . ρ is the volume charge density, C/m3 . Here, l, S, and V represent the length, surface, and volume, respectively, over which the charge is distributed. EXAMPLE 3.1 Electric field strength due to two point charges in free space. Evaluate the electric field strength for two point charges of Q1 and Q2 in vacuum. The charges are located on the x-axis and separated by a distance 2b as shown in Figure 3.7. Solution Consider point A located at (r, θ) of a spherical coordinate system. The electric field strength at point A is given by Eq. (3.13) as 1 E= 4π ǫo



Q1 Q2 i1A + 2 i2A R12 R2



(3.18)

3.1 BASIC ELECTROSTATICS IN FREE SPACE

39

y ir

i2A

i1A A R1

Q1

r

R2

O

b

b

Q2

x

Figure 3.7. Two point charges Q1 and Q2 equally spaced from the origin.

As the electric field strength is a vector, we need to evaluate the components of E in r and θ directions. The components are given by Er and Eθ , respectively, where (3.19)

E = Er ir + Eθ iθ

Here, ir and iθ are the unit vectors in r and θ directions, respectively. Let us evaluate the r-component of the electric field strength, Er . One can use Eq. (3.18) together with Eq. (3.19) to give Er ir + Eθ iθ =

1 4π ǫo



Q1 Q2 i + 2 i2A 2 1A R1 R2



(3.20)

Performing dot product on the terms in Eq. (3.20) with ir gives 1 Er = 4π ǫo



Q1 Q2 i · ir + 2 i2A · ir 2 1A R1 R2



(3.21)

From the geometry of Figure 3.7, one can write i1A · ir = cos α

(3.22)

i2A · ir = cos β

(3.23)

b sin θ R1

(3.24)

b sin(π − θ ) R2

(3.25)

sin α = sin β =

R12 = r 2 + b2 − 2rb cos θ

(3.26)

R22 = r 2 + b2 − 2rb cos(π − θ )

(3.27)

40

ELECTROSTATICS

Making use of Eqs. (3.22)–(3.27), Eq. (3.21) becomes 

 2 2 2 sin2 θ 2 sin2 θ R − b R − b Q Q 2 1  1 1 2  Er = +   3 3 4π ǫo R1 R2

(3.28)

where R1 and R2 are provided by Eqs. (3.26) and (3.27), respectively. The θ-component, Eθ , can be evaluated by performing dot product of the terms in Eq. (3.20) by iθ and recognizing that i1A · iθ = cos i2A · iθ = cos and using the sine theorem

π

 − α = sin α

(3.29)

 + β = −sin β

(3.30)

2

π 2

b sin θ R2

sin β =

(3.31)

to give Eθ =

b sin θ 4π ǫo



Q1 Q2 − 3 R13 R2



(3.32)

The magnitude of E is given by E=



Er2 + Eθ2

(3.33)

Equations (3.28) and (3.32) provide the components of the electric field strength, E, at point A. Let us consider the special case of Q1 = +Q and Q2 = −Q. Let the separation distance 2b be very small, i.e., R1 ≫ b, R2 ≫ b, and r ≫ b. Neglecting the small terms of the order of (b/r), we can write from Eqs. (3.26) and (3.27) R1 ≃ R2 ≃ r and (R2 − R1 ) ≃ 2b cos θ The r-component of E as given by Eq. (3.28) for this case gives Q Er = 4π ǫo



   1 (R2 − R1 )(R2 + R1 ) 1 Q − 2 = 4π ǫo R12 R2 R12 R22

or Er =

2(2b)Q cos θ 4π ǫo r 3

(3.34)

3.1 BASIC ELECTROSTATICS IN FREE SPACE

The θ-component of E as provided by Eq. (3.32) gives   Qb sin θ 1 1 Q(2b) sin θ Eθ = + 3 = 4π ǫo 4π ǫo r 3 R13 R2

41

(3.35)

Making use of Eq. (3.19), the electric field strength is given by E=

Q(2b) [2 cos θ ir + sin θ iθ ] 4π ǫo r 3

(3.36)

This is the field strength for two point charges +Q and −Q located on the x-axis and separated by a distance 2b. A system of two point charges of +Q and −Q separated by a very small distance of 2b constitutes what is called a point-like dipole. It is interesting to note that two charges of opposite signs at very small separations produce an electric field. Equation (3.36) represents the electric field strength generated by a dipole in free space. We will discuss dipoles at a later stage as they are important in non-conducting materials, i.e., dielectric materials. EXAMPLE 3.2 Electric field strength in free space due to a charged spherical shell. Evaluate the electric field strength at a distance z from the center of a spherical surface having radius a placed in vacuum. The surface charge density of the spherical shell is qs (C/m2 ). The geometry is shown in Figure 3.8. Solution Let us start with Eq. (3.16) which describes the electric field strength at a location in space due to a surface charge distribution  1 qs i E= dS (3.37) 4π ǫo S R 2 Let point A, located at a distance z from the origin, represent the position where the electric field strength is to be evaluated. For the problem at hand, the unit vector ir is along OB, unit vector i is along BA, and iz is the unit vector along the z-coordinate. From the geometry of Figure 3.8, one can write i · iz = cos α

(3.38)

R 2 = a 2 + z2 − 2az cos θ

(3.39)

cos α =

(z − a cos θ ) (a 2 + z2 − 2az cos θ )1/2

(3.40)

and dS = a 2 sin θ dφ dθ

(3.41)

42

ELECTROSTATICS

z iz

i

A

R ir

B dS a x

Figure 3.8. Spherical surface with a specified surface charge density qs .

Recognizing that Ez = E · iz and making use of Eqs. (3.38)–(3.41), Eq. (3.37) becomes  π  2π 2 qs a (z − a cos θ ) sin θ dφ dθ Ez = (3.42) 4π ǫo 0 (a 2 + z2 − 2az cos θ )3/2 0 Integration over φ leads to a 2 qs Ez = 2ǫo



π

0

(z − a cos θ ) sin θ dθ (a 2 + z2 − 2az cos θ )3/2

(3.43)

Making the substitution of cos θ = t, Eq. (3.43) can then be integrated to yield Ez =

a 2 qs ǫo z 2

for |z| > a

(3.44)

outside the spherical shell, and Ez = 0 inside the spherical shell.

for |z| < a

(3.45)

3.1 BASIC ELECTROSTATICS IN FREE SPACE

43

The outer electric field strength decays with the square of the distance from the center and it is zero inside the spherical shell.

3.1.3 The Gauss Law The Gauss law relates the electric field strength flux through a closed surface to the enclosed charge. To derive the Gauss law, let us consider a point charge Q located in some arbitrary volume, V , bounded by a surface S as shown in Figure 3.9. Let n be a unit outward normal vector to the bounding surface. The electric field strength at the element of surface dS due to the charge Q is given by E=

Qi 4π ǫo R 2

(3.46)

where the unit vector i is directed from the point charge to the surface element dS and R is the distance between them. Performing dot product with n dS and integrating over the bounding surface S, Eq. (3.46) gives   (i · n)Q dS (3.47) (E · n) dS = 2 S 4π ǫo R S Recognizing that the element of solid angle d is given by (i · n)dS/R 2 , Eq. (3.47) becomes   4π 1 Qd (3.48) (E · n) dS = 4π ǫo 0 S

Figure 3.9. Surface enclosing a point charge Q.

44

ELECTROSTATICS

Upon integration, Eq. (3.48) gives 

Q ǫo

(E · n) dS = S

(3.49)

Equation (3.49) states that the electric field strength flux through the closed surface S is proportional to the charge enclosed by the surface. Such a relationship is independent of the surface shape or the position of the charge within the surface enclosure. Making use of the superposition principle, one can generalize Eq. (3.49) so that the point charge Q is the sum of all point charges within the surface enclosure S. Equation (3.49) is the integral form of the Gauss law or theorem. It simply states that if there is no net charge within the enclosure surface S, then the electric field flux is zero. Based on the Gauss law one should expect a null electric flux arising from, say, a piece of polymer where its net charge is zero. The differential form of the Gauss law can be derived quite readily using the divergence theorem, which states 



(∇ · E) dV =

V

(E · n) dS

(3.50)

S

Here, volume V is enclosed by surface S. One can express the charge density ρ within the volume V in terms of the total charge Q as Q=



(3.51)

ρ dV

V

From Eqs. (3.49) to (3.51), one can write 

(∇ · E) dV =

V

1 ǫo



ρ dV

(3.52)

S

As volume V is arbitrary, Eq. (3.52) reduces to ∇ ·E=

ρ ǫo

(3.53)

This is the differential expression of the Gauss law.

3.1.4

Electric Potential

It is of interest to investigate certain properties of the electric field strength, E, that would facilitate the solution of problems involving point and distributed charges. Without loss of generality, let us consider a single point charge located at the origin of the system of coordinates.

3.1 BASIC ELECTROSTATICS IN FREE SPACE

45

Utilizing Eq. (3.14), one can write E=

Qr 4π ǫo r 3

(3.54)

where r = rir , following Eq. (3.5). A useful vector identity is given by (3.55)

∇ × (f G) = f (∇ × G) + ∇f × G

where f is a scalar function and G is a vector function. We may represent Eq. (3.54) as a product of a scalar and a vector function by writing f =

Q 4π ǫo r 3

(3.56)

and (3.57)

G=r

Taking the curl (∇×) of Eq. (3.54) and utilizing the vector identity, Eq. (3.55), we obtain     Q Q ∇ ×E= (∇ × r) + ∇ ×r (3.58) 4π ǫo r 3 4π ǫo r 3 Performing the differentiation in Eq. (3.58) provides ∇ ×E=



Q 4π ǫo r 3



(∇ × r) −



3Q 4π ǫo r 4



∇r × r

(3.59)

Recognizing that ∇=

∂ ∂ ∂ iy + iz , ix + ∂x ∂y ∂z

r = xix + yiy + ziz

(3.60) (3.61)

and utilizing the definition of a curl, one can show that ∇ ×r =0

(3.62)

∇r × r = 0

(3.63)

and These identities, when substituted in Eq. (3.59), yield ∇ ×E=0 Equation (3.64) states that the curl of the electric field strength is zero.

(3.64)

46

ELECTROSTATICS

Such a property would lend the name irrotational field to the electric field strength E and would allow it to possess certain properties that are useful in electrostatics. Some of these properties are (Griffiths, 1989): (a) (b)

2

1

E · dt is independent of path for given end points 1 and 2.

E · dt = 0 for any closed path. This is a “conservative” property.

(c) E = −∇ψ where ψ is some scalar function. Here t is a unit vector along a line path traced on a surface. The property under (c) is of much interest. It implies that the electric field strength can be calculated as a gradient of a single scalar function, the potential function, which we will designate as ψ. We write this fundamental relationship between the irrotational field and the scalar potential as (3.65)

E = −∇ψ

By using the scalar function ψ, one can reduce a vectorial problem to a scalar one. Consequently, in many cases it is easier to work with the electric potential, ψ, rather than with the electric field strength, E. Let us find some physical meaning to the electric potential, ψ. The total work, W12 , required to move a point test charge Q a finite distance between two points 1 and 2 in an electric field E, as shown in Figure 3.10, is given by W12 = −



2

(3.66)

QE · dt

1

where we note that QE is the force according to Eq. (3.12) and t is the path along which the charge Q travels. From Eqs. (3.65) and (3.66), one can write 

2

Q∇ψ · dt

(3.67)

W12 = Q(ψ2 − ψ1 )

(3.68)

W12 =

1

and obtain

n

t 1

2

Figure 3.10. A point charge moving from location 1 to 2 along path t whose normal is n.

47

3.1 BASIC ELECTROSTATICS IN FREE SPACE

Equation (3.68) shows that the difference in the electric potential between two points is the work done to move a unit test (and point) charge in an electric field, E. This equation also indicates that the work is independent of the path taken between points 1 and 2. Such a property is characteristic of a conservative field. Let us now derive an expression for the electric potential due to a point charge located at the origin. Making use of the electric field strength expression for a point charge given by Eq. (3.14) and utilizing the relationship between E and ψ from Eq. (3.65) one can write Q ir (3.69) ∇ψ = − 4π ǫo r 2 As the variation of ψ is in the radial direction only, Eq. (3.69) simplifies to dψ Q =− dr 4π ǫo r 2

(3.70)

Integration of Eq. (3.70) provides Q ψ = ψ1 + 4π ǫo



1 1 − r r1



(3.71)

where ψ = ψ1 at r = r1 . Here, we can think of ψ1 being a reference value for the electric potential at a radial distance r1 . By taking the reference location at infinity, and letting ψ1 to be zero at infinity, Eq. (3.71) gives the electric potential for a point charge located at the origin. Q (3.72) ψ= 4π ǫo r As indicated by Eq. (3.72), the potential due to a point charge decays with r rather than r 2 as was the case for the electric field strength. A generalization for the electric potential at location A due to a point charge Q1 , shown in Figure 3.11, is given by ψ=

Q1 4π ǫo |rA − r1 |

(3.73)

y Q1 A

r1 rA

x z Figure 3.11. Electric potential at location A due to a point charge Q1 located at r1 .

48

ELECTROSTATICS

|rA − r1 | represents the distance between the point charge Q1 and location A. By letting |rA − r1 | = R1 , one obtains ψ=

Q1 4π ǫo R1

(3.74)

where R1 is the separation distance between point charge Q1 and point A. By invoking the superposition principle, one can write an expression for the electric potential at location A due to the presence of many point charges. n n 1  Qi 1  Qi = ψ= 4π ǫo i=1 |rA − ri | 4π ǫo i=1 Ri

(3.75)

where Ri = |rA − ri | represents the separation distance between point charge Qi and location A and ri is the position vector of charge Qi . The total number of point charges is n. Similar to the case of the electric field strength, one can write expressions for the electric potential at location A due to a line, surface, and volume carrying a charge distribution. The relevant electric potential at location A in space are expressed as: For a line charge  1 λ dl (3.76) ψ= 4π ǫo line R For a surface charge ψ=

1 4π ǫo



1 4π ǫo



(3.77)

surface

qs dS R

(3.78)

volume

ρ dV R

For a volume charge ψ=

where λ, qs , and ρ are the line, surface, and volume charge densities, respectively, as was defined previously for the electric field strength. Here, R is the distance between the charge carrying element and location A. The differential form of the electric potential can be easily derived by combining Eqs. (3.53) and (3.65) to give ∇ · (∇ψ) = −

ρ ǫo

(3.79)

or, ∇ 2ψ = −

ρ ǫo

(3.80)

3.1 BASIC ELECTROSTATICS IN FREE SPACE

49

Equation (3.80) is known as Poisson’s equation. In cases where the electric charge density within a body is absent, i.e., ρ = 0, The Poisson equation (3.80) becomes ∇ 2ψ = 0

(3.81)

Equation (3.81) is known as Laplace’s equation. Solution of either the Poisson or Laplace equations is subject to appropriate boundary conditions. Discussion on the boundary conditions will be given in Section 3.5. EXAMPLE 3.3 Electric potential in free space due to a charged spherical shell. Let us solve Example 3.2 by evaluating the electric potential at point z due to the placement of a charged spherical surface having a surface charge density of qs (C/m2 ). Solution

From Eq. (3.77) we can write, 1 ψ(z) = 4π ǫo



S

qs dS R

(3.82)

Making use of Figure 3.8, one obtains R 2 = a 2 + z2 − 2az cos θ

(cosine law)

dS = a 2 sin θ dφ dθ

(3.83) (3.84)

From Eqs. (3.83) and (3.84), Eq. (3.82) becomes ψ(z) =

qs 4π ǫo

2π

a 2 sin θ dφ dθ (a 2 + z2 − 2az cos θ )1/2

(3.85)

a 2 qs ǫo z

outside the surface

(3.86)

aqs ǫo

inside the surface

(3.87)



π

0



0

Upon integration, we obtain ψ(z) =

ψ(z) =

Let the total charge, Qs , be related to the surface charge density, qs , as Qs = 4π a 2 qs

(3.88)

Qs 4π ǫo z

(3.89)

Equation (3.86) then gives ψ(z) =

for z ≥ a.

50

ELECTROSTATICS

At the shell’s surface z = a, one obtains ψs =

Qs 4π ǫo a

(3.90)

Equation (3.90) relates the total charge to the surface potential at the spherical shell. In terms of surface charge density distribution, we obtain qs =

ǫo ψs a

(3.91)

Clearly, one would have obtained the same results by making use of the solution of the field E obtained in Example 3.2.

3.2 SUMMARY OF ELECTROSTATIC EQUATIONS IN FREE SPACE The pertinent governing electrostatic equations in free space are given below in their integral and differential forms. The symbols were previously defined and they are also provided in the nomenclature at the end of this chapter. 3.2.1

Integral Form

The relevant electrostatic equations in free space are listed in Tables 3.1 and 3.2. In Table 3.1, i1A is the unit vector along point charge Q1 to point A, RiA is the displacement vector defined as rA − ri , rA is the position vector of location A, and ri is the position vector of point charge i. 3.2.2

Differential Form

The relevant electrostatic equations in free space in their differential form are provided in Table 3.3, where E = −∇ψ and ∇ × E = 0. In order to use the differential form of the electrostatic equations, we need to define the boundary conditions pertinent to a given physical situation. A discussion on the TABLE 3.1. Integral Forms of the Electrostatic Equations in Free Space for Discrete Point Charges.

Point charge Q1 at origin Point charge Q1 at r1 Multiple point charges

Electric Field Strength, E

Electric Potential, ψ

Q1 ir 4π ǫo r 2 Q1 i1A 4π ǫo |R1A |2 1
Qi iiA 4π ǫo |RiA |2

Q1 4π ǫo r Q1 4π ǫo |R1A | 1
Qi 4π ǫo |RiA |

3.3 ELECTROSTATIC CLASSIFICATION OF MATERIALS

51

TABLE 3.2. Integral Forms of the Electrostatic Equations in Free Space for Distributed Charges.

Line charge Surface charge Volume charge

Electric Field Strength, E  iλ 1 dl 4π ǫo R2  iqs 1 dS 4π ǫo R2  iρ 1 dV 4π ǫo R2

Electric Potential, ψ 1  λ dl 4π ǫo R  qs 1 dS 4π ǫo R  1 ρ dV 4π ǫo R

TABLE 3.3. Differential Forms of the Electrostatic Equations in Free Space. Electric field strength, E Electric potential, ψ

ρ ǫo ρ ∇2ψ = − ǫo ∇ ·E=

boundary conditions will follow at a later stage after we have discussed electrostatics in real media, i.e., in presence of free charge. The electrostatic equations derived so far are applicable to charges in free space, i.e., vacuum. When free space is replaced by a real medium, the electrostatic equations for free space need to be modified. First, we shall discuss briefly the electrical classification of materials and then show how to modify the electrostatic equations in real media.

3.3 ELECTROSTATIC CLASSIFICATION OF MATERIALS Interaction of electrical potentials and fields with physical materials gives rise to different effects depending on how the material responds to the electrical parameters. From the perspective of classical electrostatics, it is possible to broadly differentiate matter into two classes, namely, conductors and dielectrics or insulators. Conductors are materials that have a large supply of free charge, which can freely move from one region of the material to another under the influence of an externally imposed electric potential difference across the material. Free charge can exist in different forms in different materials. For instance, in conducting metals (such as copper or gold), the free charge carriers are the mobile electrons. In aqueous electrolyte solutions (for instance, a solution of sodium chloride in water), the dissociated ions are the free charge carriers. When an electrical potential difference is applied across such a conducting material, the free charges will move to the regions of different potentials depending on the type of charge they carry. For instance, in a metallic conductor, the mobile electrons (negative charge) will be transported to the positive potential. If the metallic conductor is also connected to a source of electrons (negative potential), then

52

ELECTROSTATICS

electrons will move into the conductor to fill in the voids left by the depleting electrons. Such a process will set up an electric current (flow of electrons or charge) through the conducting material. This process is called electrical conduction. The extent of current flow under an applied potential difference across a conducting material is governed by the electric conductivity of the material. The inverse of the electric conductivity is known as the electric resistivity. Low conductivity (or high resistivity) signifies a poor conductor or an insulator. A poor conductor either does not possess sufficient amount of free charges, or is unable to transport the free charges easily. The voltage-current characteristics of conducting materials are related by Ohm’s law V =

I = IR K

where V is the applied potential difference (Volt) across the material, I is the current (Ampere) flowing through the material, K is the electric conductance of the material (Ampere/Volt = Siemens), and R = K −1 is the electric resistance of the material, (Volt/Ampere = Ohm). Dielectrics, on the other hand, are materials that have a capacity of storing charge across their surfaces when a potential difference is applied across such a material. Ideally, dielectrics are materials with no conductance or infinite resistance, implying that these materials do not have free or mobile charges. Consequently, there will be no direct current flow through perfect dielectric materials. When a potential difference is set up across such dielectric materials, the material tends to develop opposite charges on its two surfaces. The mechanism of accumulation of charges at the two ends of a dielectric is called polarization. Dielectric materials, although devoid of free or mobile charges, may contain bound charge carriers which can move or reposition in an atomic or molecular scale. Such repositioning allows dielectric molecules or atoms to act as electric dipoles. These dipoles can orient themselves under the influence of an electric field. The alignment of dipoles under the influence of an external electric field is termed as electric polarization. Let us consider a dielectric material placed in an external electric field. To simplify the discussion, let us assume that the dielectric material is made up of “perfect” neutral spherical atoms. What is meant by a “perfect” neutral spherical atom is that there is a positively charged core, i.e., a nucleus, and a negatively charged electron cloud surrounding the nucleus. Collectively, the nucleus carries a charge +Q and the electrons a charge of −Q, although both types of charges are centered at the same location. For such a “perfect” atom, its net charge is zero, and there is no dipole moment in absence of an external electric field. However, when an external electric field is present, the atom can acquire a dipole moment. It should be recalled from Example 3.1, that a pair of point charges +Q and −Q separated by a small distance is called an electric dipole and such a charge configuration would normally produce an electric field. Upon the placement of such a dielectric material in an external electric field, there will be a charge separation within each atom as shown in Figure 3.12(a). The electrons within an atom are shifted slightly from their original position with respect to the nucleus. Such a shift in position would transform each atom into a dipole,

3.3 ELECTROSTATIC CLASSIFICATION OF MATERIALS

53

Figure 3.12. Polarization of a dielectric material in presence of an electric field. (a) Deformational polarization of neutral spherical atoms. (b) Orientational polarization of a permanent dipole molecule. In both cases, there is no polarization charge in absence of an external electric field. In presence of an external electric field, the polarization charge is set up due to the two mechanisms shown.

i.e., two charges of +Q and −Q separated by a finite but small distance. We refer to these atoms as being polarized. The mechanism of polarization of neutral atoms in the above manner is often termed as “deformational polarization”. What is of interest is that the induced charge separation within the atoms manifests itself as a “polarization” charge at the extremities of the dielectric material. Such a polarization charge is called “bound charge”. In other words, when a dielectric material is placed in an external electric field, bound charges (or polarization charges) appear at the surface of the dielectric medium. In electrostatics, the appearance of bound charges is of interest. Development of bound charges is not limited to dielectrics made up of neutral atoms (i.e., non-polarized atoms or molecules). Bound charges are also manifested when dielectrics made up of molecules that are already polarized (i.e., they are dipoles in absence of any external electric field, better known as permanent dipoles) are subjected to an external field. In this case, the bound charge appearance would be due to the rotation of the molecules as well as to charge separation. The polarization caused by re-orientation of a molecule with a permanent dipole in an externally imposed electric field is often referred to as “orientational polarization”, and the mechanism is depicted in Figure 3.12(b). Some examples of dielectric materials are glass, silk, poly-(methyl methacrylate), water, ethyl alcohol, and air. The ability of a dielectric to accumulate charges at its outer surfaces is termed as its capacitance. The capacitance is related to the charge accumulated across the dielectric’s surfaces and the applied potential difference by the expression V =

Q C

(3.92)

where V is the potential difference across the dielectric (Volt), Q is the total accumulated charge across the dielectric (Coulomb), and C is the capacitance

54

ELECTROSTATICS

(Coulomb/Volt = Farad). The above equation may be considered as analogous to Ohm’s law for conducting materials, with the conductance replaced by capacitance and the current replaced by the accumulated charge. Let us consider the phenomenon of charging of a capacitor, and assess how different dielectric materials influence the charging process. For this, we will refer to Figure 3.13. Figure 3.13(a) depicts two parallel plates being held at different potentials V1 and V2 , which are separated by a distance d. The intervening medium is vacuum. In this case, applying the Gauss law, it can be shown that the electric field E0 between the plates will be related to the charge on the plates, Q, through E0 =

Q ǫo A

(3.93)

Here, Q is the charge on one of the plates, ǫo is the permittivity of vacuum, and A is the cross-section area of the plate, implying that the charge density on the plate is q = Q/A. Note that in this type of parallel plate capacitors, if V1 > V2 , then the plate on the left will acquire a charge of +Q and the plate on the right will acquire a charge −Q. We also note that the field can be expressed as E0 = V /d, where V = V1 − V2 . Therefore, Eq. (3.93) can be written as V =

Q ǫo A/d

(3.94)

Comparing Eqs. (3.92) and (3.94) it may be noted that ǫo A/d is the capacitance of the vacuum between the plates. Consider now that a dielectric medium is placed between the two plates instead of the vacuum as shown in Figure 3.13(b). The medium will be polarized in presence of

Figure 3.13. Modification of electric field by a dielectric material. (a) Two charged parallel plates in vacuum. (b) Same plates with a dielectric material in between.

3.3 ELECTROSTATIC CLASSIFICATION OF MATERIALS

55

the applied electric field, and as a consequence, the surface of the medium adjacent to the left plate will acquire a negative charge −Qm , while the surface of the medium adjacent to the right hand side plate will acquire a positive charge +Qm . These charges will oppose the original charges of the plates, and the magnitude of the net charge will be Q − Qm at the plate-medium interface. In this case, the effective electric field, Em , between the plates through the dielectric medium will be Em =

(Q − Qm ) ǫo A

(3.95)

The effective field is an outcome of the counteracting field due to the polarized medium opposing the original electric field in vacuum, E0 . Writing the field as Em = V /d, Eq. (3.95) can be rearranged as V =

(Q − Qm ) Q Q = = A Q A A ǫo ǫo ǫo ǫr d (Q − Qm ) d d

(3.96)

where ǫr =

Q (Q − Qm )

The dimensionless parameter ǫr is termed as the relative permittivity of the medium, or the dielectric constant of the medium, and signifies how effectively the medium polarizes in an applied electric field compared to vacuum. If the medium polarizes very poorly, then Qm → 0 and ǫr → 1. However, if the medium is highly polarizable, the polarized charge Qm is significant, and hence, the dielectric constant of the medium becomes a large number. The capacitance of the dielectric material is expressed as C = ǫo ǫr

A d

(3.97)

Comparing Eqs. (3.94) and (3.96), it becomes clear that when the same potential difference V is applied between the parallel plates, the charge accumulation across the plates when the intervening medium is a dielectric material is increased by a factor ǫr compared to the charge accumulation in vacuum. The dielectric constant, ǫr of a dielectric material is a positive quantity with a minimum value of 1 corresponding to vacuum. The dielectric constant is often quite large for highly polarizable materials. For instance, water has a dielectric constant of 78.54 at 25 ◦ C. When describing conducting materials, we briefly mentioned that aqueous electrolyte solutions can act as conductors. Electrolyte solutions are peculiar in that they contain mobile charges in the form of ions suspended in a dielectric solvent. A simple example is the case of water containing dissolved common salt, NaCl. Upon dissolution, the salt dissociates into sodium, Na+ , and chloride, Cl− , ions, which can freely move around in the water. Such an electrolyte solution can be considered as

56

ELECTROSTATICS

an ionic conductor. By placing two electrodes connected to a battery in an aqueous solution containing salts, an electric current flow can be established between the two electrodes. The current is due to the movement of the salt ions. In the case of sodium chloride salt, it is the movement of the Na+ and Cl− ions that gives rise to the electric current between the electrodes. The mechanism of current flow in ionic conductors is quite different from that in metallic or electronic conductors. Ionic conductors play a major role in biological functions and in many aspects of electrostatics, particularly electrokinetic phenomena. This notion of an ionic conductor, or simply a dielectric material containing free ions, is a prerequisite for dealing with electrostatic double layers, which will be discussed in Chapter 5. 3.4 BASIC ELECTROSTATICS IN DIELECTRICS The basic electrostatic equations in free space were discussed in Section 3.1. We need now to extend our discussion to include dielectric materials. In this section we will derive the relevant electrostatic equations for a dielectric medium. The molecules of a dielectric material constitute dipoles in the presence of an electric field. A dipole comprises two equal and opposite charges, +Q and −Q, separated by a distance d. A dipole moment is defined as Qd, a vector quantity, where d is the vector orientation between the two charges as shown in Figure 3.14 (a vector describing the average orientation of the dipole and the charge separation distance). For bulk phases, the polarization dipole moment or polarization field strength P is given by P = NQd

(3.98)

where N is the number of dipoles per unit volume. For most dielectric materials, when the electric field is not too strong, the polarization is directly proportional to the applied field, and one can write P = χ ǫo E

(3.99)

A material that obeys the proportionality expressed by Eq. (3.99) is called a linearly polarized dielectric material or simply a linear dielectric. Here χ is the electric susceptibility of the dielectric medium and it is dimensionless. Through complex averaging arguments, one can write ∇ · P = −ρp

d

Figure 3.14. Two charges separated by a distance d.

(3.100)

3.4 BASIC ELECTROSTATICS IN DIELECTRICS

57

where ρp is the volumetric polarization (or bound) charge density. Equation (3.100) simply states that the polarization field P is related to the polarization charge. Within a dielectric material, the total volumetric charge density is made up of two types of charge densities, a polarization and a free charge density ρ = ρ p + ρf

(3.101)

For lack of better terminology, ρf is termed as free volumetric charge density. This “free” charge can consist of ions in an electrolyte solution, electrons on a conductor, or ions embedded in the dielectric material. In other words, the term free charge, ρf , constitutes any charge that is not the result of polarization. Combining the definition of total charge density provided by Eq. (3.101) with the Gauss law, Eq. (3.53), gives ∇ ·E=

1 (ρp + ρf ) ǫo

(3.102)

Substituting for the polarization charge, ρp from Eq. (3.100), the divergence of the electric field becomes 1 ∇ · E = (−∇ · P + ρf ) (3.103) ǫo leading to ∇ · (ǫo E + P) = ρf

(3.104)

The term (ǫo E + P) is usually designated as the electric displacement vector, D. Equation (3.104) can then be written in the format of the Gauss law as ∇ · D = ρf

(3.105)

Utilizing the divergence theorem, Eq. (3.105) can be written in the integral form as  D · n dS = Qf (3.106) S

where Qf is the total free charge enclosed by surface S. Equations (3.105) and (3.106) are part of the Maxwell set of electrostatic equations for dielectrics. We will now make use of the equation relating the polarization dipole moment to the electric field as given by Eq. (3.99). Combining Eqs. (3.99) and (3.104) leads to ∇ · [ǫo (1 + χ )E] = ρf

(3.107)

ǫ = ǫo (1 + χ )

(3.108)

Let We will call ǫ the permittivity of the material. Combining Eqs. (3.105), (3.107), and (3.108) gives D = ǫE

(3.109)

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ELECTROSTATICS

In free space, there is no material to be polarized, and therefore χ is zero. Here, the permittivity simply reduces to ǫo , which is the permittivity of free space. Consequently, the definition of ǫ provided by Eq. (3.108) is consistent. Combining Eqs. (3.107) and (3.108) leads to ∇ · (ǫE) = ρf

(3.110)

For constant permittivity, ǫ, Eq. (3.110) gives ǫ∇ · E = ρf

(3.111)

Equation (3.111) is Maxwell’s equation for a dielectric material. If one is to define (1 + χ ) as ǫr = (1 + χ )

(3.112)

then the equivalent equations to Eqs. (3.110) and (3.111) become, respectively, ǫo ∇ · (ǫr E) = ρf

(3.113)

ǫo ǫr ∇ · E = ρf

(3.114)

and for ǫr being constant The term ǫr is called the dielectric constant or relative permittivity of the dielectric material and is a dimensionless number. The use of (ǫo ǫr ) instead of ǫ is convenient as ǫr for different dielectric materials varies within a numerical range that is easily remembered. The minimum value of ǫr is unity for vacuum. Its values range from near unity for most gases, 2–4 for oil crudes, 15–35 for alcohols, and about 80 for water. The use of relative permittivity (or dielectric constant) is akin to the use of the specific gravity (or relative density) of substances using the density of water as a reference. If one is to strictly adhere to SI convention, then the use of a material permittivity ǫ is more appropriate. In some literature the symbols ǫr and ǫ are used interchangeably. Making use of the potential relationship given by Eq. (3.65), Eq. (3.110) can be written as ∇ · (ǫ∇ψ) = −ρf

(3.115)

For constant permittivity, one can write ∇ 2ψ = −

ρf ǫ

(3.116)

Equation (3.116) represents the Poisson’s equation for the electric potential distribution in a dielectric material. In situations where there is no free charge, one can set ρf to zero, and Eq. (3.116) becomes ∇ 2ψ = 0

(3.117)

Equation (3.117) is referred to as Laplace’s equation, which describes the electric potential distribution in a material having no free charge.

3.4 BASIC ELECTROSTATICS IN DIELECTRICS

59

EXAMPLE 3.4 Electric Potential Distribution. Obtain an expression for the electric potential distribution for a point charge Q in a dielectric in the absence of free charge. Solution

In the absence of free charge, Laplace’s equation is applicable, where ∇ 2ψ = 0

(3.118)

Since there is no θ- and φ-dependence in a spherical coordinate system, Eq. (3.118) becomes   1 d 2 dψ r =0 (3.119) r 2 dr dr the solution of which is A1 dψ = 2 dr r

(3.120)

and ψ =−

A1 + A2 r

(3.121)

where A1 and A2 are integration constants. The constant A2 can be evaluated using the condition that at large distances, i.e., as r → ∞, the effect of the point charge is negligible and consequently, ψ → 0. This condition implies that A2 = 0. In order to evaluate A1 we need to make use of the charge Q. Consider Eq. (3.111) where ǫ∇ · E = ρf Consider a spherical shell enclosing the point charge. Applying the divergence theorem yields    ρf dV (3.122) (∇ · E) dV = (E · ir ) dS = V ǫ S V Since E · ir = Er = −dψ/dr, the above equation becomes −



S

dψ dS = − dr

  S

A1 r2



dS =

Q ǫ

(3.123)

 where the term V ρf dV represents the charge Q. Note that the charge Q is the free charge in the spherical volume. As the surface element dS = r 2 dφ, Eq. (3.123) becomes  4π Q A1 dφ = − (3.124) ǫ 0

60

ELECTROSTATICS

leading to A1 = −

Q 4π ǫ

Therefore, the potential is given as Q 4π ǫr

ψ=

(3.125)

The above equation for the potential due to a point charge in a dielectric is similar to that given in free space, Eq. (3.72) with ǫo being replaced by ǫ. If one replaces the point charge by a spherical shell of radius a having a uniform surface charge distribution of qs that is equivalent to a total free surface charge of Qs , then the equivalent to Eq. (3.125) becomes ψs =

Qs 4π ǫa

(3.126)

The above equation can be compared to the case of free space as provided by Eq. (3.90). The analysis conducted in this example and in the previous Example 3.3, clearly shows how the surface potential and surface charge density can be related to each other. It should be noted that Eq. (3.126) is also an expression given to a capacitor, where 4π ǫa is known as the capacity of the spherical shell Qs = 4π ǫa = C ψs

(3.127)

where C is the capacitance. Now, let us return to the force exerted by two free point charges Q1 and Q2 in a dielectric medium. Making use of Eq. (3.1) the equivalent expression for Coulomb’s law in a dielectric is given by F =

Q1 Q2 4π ǫR 2

(3.128)

Allowing ourselves to write ǫ = ǫo ǫr , Eq. (3.128) becomes F =

Q1 Q2 4π ǫo ǫr R 2

(3.129)

where R is the distance separating the two point charges. Comparing the force between the two point charges in free space as was given by Eq. (3.1) and that in a dielectric medium of permittivity ǫo ǫr , as given by Eq. (3.129), it is clear that the force between the two charges is reduced by a factor ǫr due to the presence of the dielectric material in which the point charges are located.

61

3.4 BASIC ELECTROSTATICS IN DIELECTRICS

One can think of the dielectric constant or relative permeability, ǫr , in terms of a ratio which is given by ǫr =

Force between two charges in free space Force between the same charges, at the same separation, in a dielectric medium

ǫr is a dimensionless quantity with ǫr > 1. Typical values of the dielectric constant are provided in Table 3.4. For gases at atmospheric pressure, ǫr ≈ 1, and ǫr increases with pressure. For example, air at 1 atm has an ǫr value of about 1.0006 whereas at 100 atm, ǫr = 1.055. The dielectric constant for water is about 80. The dielectric constant is a function of temperature and for the case of water ǫr variation with temperature is provided in Table 3.5.

TABLE 3.4. Dielectric Constants* ǫr of Some Common Liquids and Solids at 25 ◦ C. Compound Hydrogen Bonding Methyl formamide Formamide Hydrogen fluoride

ǫr

Ethylene glycol Methanol Ethanol n-Propanol Ammonia Acetic acid

HCONHCH3 HCONH2 HF (at 0 ◦ C) H2 O D2 O HCOOH (at 16 ◦ C) C2 H4 (OH)2 CH3 OH C2 H5 OH C3 H7 OH NH3 CH3 COOH

Non-Hydrogen Bonding Acetone Chloroform Benzene Carbon tetrachloride Cyclohexane Dodecane Hexane

(CH3 )2 CO CHCl3 C 6 H6 CCl4 C6 H12 C12 H26 C6 H14

Water Heavy water Formic acid

*Taken from Israelachvili (1985).

182.4 109.5 84.0 78.5 77.9 58.5 40.7 32.6 24.3 20.2 16.9 6.2

20.7 4.8 2.3 2.2 2.0 2.0 1.9

Compound

ǫr

Polymers Nylon Fluorocarbons Polycarbonate Polystyrene PTFE

3.7–4.2 2.1–3.6 3.0 2.4 2.0

Glasses Fused quartz SiO2 Soda glass Borosilicate glass

3.8 7.0 4.5

Crystalline Solids Diamond (carbon) Quartz SiO2 Mica Sodium chloride Alumina Al2 O3

5.7 4.5 5.4–7.0 6.0 8.5

Miscellaneous Paraffin (liq.) Paraffin wax (solid) Silicone oil Helium (liq. at 2-3 K) Water (liq. at 0 ◦ C) Water (ice at 0 ◦ C) Air (dry)

2.2 2.2 2.8 1.055 87.9 91.6–106.4 1.00054

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ELECTROSTATICS

TABLE 3.5. Variation of the Dielectric Constant of Water with Temperature. Temperature, ◦ C 0 10 20 25 30 40 50 60 70 80 90 100

ǫr , Literature*

ǫr , Curve Fit†

87.90 83.96 80.20 78.54 76.60 73.17 69.88 66.73 63.73 60.86 58.12 55.51

87.9 84.0 80.2 78.5 76.6 73.2 69.9 66.7 63.7 60.8 58.1 55.5

*Adapted from Archer and Wang (1990). †Fitting equation: ǫr = 87.86 − 0.3963T + 7.306 × 10−4 T 2 , where T is in ◦ C.

In summary, Eq. (3.116) is referred to as Poisson’s equation that describes the electric potential in a dielectric. It is one of the fundamental equations to be used for the evaluation of the potential ψ in an electrolyte solution (e.g., NaCl in water). Equation (3.116) is therefore the relevant equation to be used in solving problems involving electrostatics in dielectric materials. A remarkable characteristic of Eq. (3.116) is that the charge density is the “free” charge density in the material. It is not the bound or polarization charge density, which is induced through the influence of an electric field. In an electrolyte solution, say, Na2 SO4 in water, the free charges are Na+ and SO2− 4 ions that are free to move in water. In order to solve Poisson’s equation, we need to define appropriate boundary conditions. In the following section, we discuss the boundary conditions associated with dielectric materials.

3.5 BOUNDARY CONDITIONS FOR ELECTROSTATIC EQUATIONS Poisson’s equation, given by Eq. (3.116), describes the electric potential distribution in a dielectric material in the presence of free charge for a space independent permittivity ǫ. In the absence of free charge density within the interior of the material under consideration, Poisson’s equation becomes Laplace’s equation, Eq. (3.117). In order to evaluate the electric field strength for a given physical problem, boundary conditions must be specified in order to solve the relevant differential equations. We shall present here the applicable boundary conditions at an interface between two dielectric materials having permittivities of ǫ1 and ǫ2 . If one of the materials is free space, then ǫo will be used instead.

3.5 BOUNDARY CONDITIONS FOR ELECTROSTATIC EQUATIONS

C

B

t

y

63

t

a

Medium 1 x

Interface Medium 2

b a

t t

D

A

Figure 3.15. Cylindrical surface element straddling an interface separating two dielectrics.

For convenience, let us consider a cylindrical volume whose surface is S and is spanned by the closed curve C, denoted by ABCD. The tangent to C is t. The cylindrical volume straddles the interface of the two dielectric materials. We seek boundary conditions at the interface. The geometry is depicted in Figure 3.15. Recognizing that the electric field strength is conservative, one can write cf., Eq. (3.64), ∇ ×E=0

(3.130)

Making use of Stokes theorem, one can write 

(∇ × E) · n dS =



E · dt

(3.131)

c

S

where n is an outward unit normal vector to the surface S and t is an element vector along a contour C spanning the surface S. Combining Eqs. (3.130) and (3.131) leads to 

E · dt = 0

(3.132)

ABCD

Carrying out the integration along ABCD with AB and DC parallel to the y-axis and CB and DA parallel to the x-axis, one can write E2y a + E1y a − E1x b − E1y a − E2y a + E2x b = 0

(3.133)

E1x = E2x

(3.134)

leading to

where E1x and E2x are the electric field strengths in x-direction for media 1 and 2, respectively. Similarly, E1y and E2y are the electric field strengths in y-direction for media 1 and 2, respectively.

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ELECTROSTATICS

Recognizing that Eq. (3.65) gives ∂ψ1 ∂x ∂ψ2 =− ∂x

E1x = −

(3.135-a)

E2x

(3.135-b)

and using Eq. (3.134), one obtains ∂ψ2 ∂ψ1 = ∂x ∂x

(3.136)

ψ1 = ψ2 + Constant

(3.137)

Upon integration, Eq. (3.136) gives

Equation (3.137) indicates that the potential at the interface between the two media will differ by at most a constant. Referring back to Eq. (3.68), we note that the potential difference is related to the work required in moving a unit charge. Now, consider the scenario when the cylinder in Figure 3.15 is shrunk by reducing the lengths AB and DC, and take the limit when the two lines BC and AD coincide (corresponding to a → 0). This is analogous to folding the cylindrical volume into the interface such that the two circular faces of the cylinder coincide. We note that for this limiting situation, there will be no work associated with moving a charge across the interface from medium 1 to 2 (as there is no finite length over which a charge is moved). Accordingly, the constant in Eq. (3.137) vanishes across an interface, and we obtain (3.138)

ψ1 = ψ2

Equation (3.138) states that the electric potential at the interface is the same whether one approaches the interface from medium 1 or medium 2. In other words, there is continuity of electric potential. As Poisson’s equation is a second order differential equation, it requires a second boundary condition. We will now proceed to establish the second boundary condition. Poisson’s equation is given by (3.139)

ǫ∇ · E = ρf

Integrating Eq. (3.139) over a volume V and applying the divergence theorem on the volume V enclosed by a surface S, we obtain    ρf dV = (ǫE · n) dS ǫ(∇ · E) dV = V

S

V

or, 

S

(ǫE · n) dS =



ρf dV

V

The unit vector n is outward normal to the surface S.

(3.140)

3.5 BOUNDARY CONDITIONS FOR ELECTROSTATIC EQUATIONS

65

Figure 3.16. Cylindrical volume element showing the bounding surfaces S1 , S2 , S3 , and S4 . Here, the unit normal vectors n1 and n2 point toward the positive and negative y directions, respectively. Both n3 and n4 (not shown) point in the radial direction.

Consider an element volume at the interface of two dielectric media as shown in Figure 3.16. Applying Eq. (3.140) to the surface elements leads to ǫ1 E1y S1 − ǫ2 E2y S2 + ǫ1 E1r S3 + ǫ2 E2r S4 =



ρf dV

(3.141)

V

Subscripts 1 and 2 refer to the media 1 and 2, respectively, and subscripts y and r refer to the cylindrical coordinates. Setting S1 = S2 = S, and letting the depth ‘a’ approach zero, we get S3 = S4 = 0, and Eq. (3.141) becomes ǫ1 E1y − ǫ2 E2y =

1 S



ρf dV = qsf

(3.142)

V

 We note that the term V ρf dV /S represents the free charge per unit area at the interface, or the free surface charge density at the interface, qsf . One can write Eq. (3.142) in terms of the electric potential, ψ, as −ǫ1

∂ψ2 ∂ψ1 + ǫ2 = qsf ∂y ∂y

(3.143)

In the absence of any free charge at the interface, i.e., qsf = 0, Eq. (3.143) becomes ǫ1

∂ψ2 ∂ψ1 = ǫ2 ∂y ∂y

(3.144)

In other words, for a zero surface charge density, the normal component, Dn = D · n, of the electric displacement vector, D, given by Eqs. (3.105) and (3.109) is continuous

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ELECTROSTATICS

n1

n2

Interface

Figure 3.17. Volume element straddling an arbitrary interface where “a” of Figure 3.16 approaches zero.

across an interface. However, since the dielectric permittivities of the two media separated by the interface are different (ǫ1 = ǫ2 ), the electric field, E, is discontinuous across the interface. Let us now derive a different form of Eq. (3.143) in terms of unit outward normal vectors applicable to interfaces of any arbitrary shape. Applying Eq. (3.140) to the element of Figure 3.17 and allowing a → 0, one can write  1 [ǫE · n]1 + [ǫE · n]2 = ρf dV = qsf (3.145) S V where n1 and n2 are the outward unit normal vectors acting on S1 and S2 , respectively, as shown in Figure 3.16. It is more convenient to express Eq. (3.145) in terms of the normal derivatives of the potential, which is given by −E · n = ∇ψ · n =

∂ψ ∂n

This renders Eq. (3.145) as ǫ1

∂ψ2 ∂ψ1 + ǫ2 = −qsf ∂n1 ∂n2

(3.146)

Here n1 and n2 are the coordinates normal to a surface in directions n1 and n2 , respectively. For example, in Cartesian coordinates, n1 can represent the x-coordinate and n1 becomes ix . Equations (3.138) and (3.146) are the necessary boundary conditions at an interface separating two media having different permittivities. They apply at any surface, flat or curved, and whether there is a charge or not. These boundary conditions are simply a statement of the Gauss law and the fact that the electric field is irrotational i.e., ∇ × E = 0. It should be pointed out that the free surface charge density at the interface is not an acquired state due to the imposition of the external electric field as is the case for the polarization charge density. It is a surface charge that already exists prior to the

3.5 BOUNDARY CONDITIONS FOR ELECTROSTATIC EQUATIONS

67

introduction of the external field. For instance, when a rubber ball is rubbed against a woolen cloth, a surface charge is acquired by the ball. In the process of applying the boundary conditions, this acquired “free surface charge” must be included in the analysis for the boundary conditions when the rubber ball is in an external electric field. This in spite of the fact that there might not be any free charge within the interior of the rubber ball; a case that would allow the use of Laplace’s equation to describe the potential inside the rubber ball. We have already discussed the relevant electrostatic equations for a dielectric medium. We will now briefly discuss polarization bound charges. As mentioned earlier, when a dielectric material is placed in an electric field, a polarization charge is developed at the interface. To derive the necessary boundary conditions involving polarization charges, we will utilize the derivations previously applied to the free charges. Combining Eqs. (3.101), (3.102), and (3.110), and assuming space-independent permittivity inside the dielectric material, we obtain (ǫ − ǫo )∇ · E = −ρp

(3.147)

Application of the divergence theorem over volume V enclosed by a surface S leads to   (3.148) (ǫ − ǫo )(E · n) dS = − ρp dV V

S

By analogy to the analysis employed for Eqs. (3.140), (3.145), and (3.146), one can write ∂ψ2 ∂ψ1 + (ǫ2 − ǫo ) = qsp (3.149) (ǫ1 − ǫo ) ∂n1 ∂n2 where qsp is the polarization surface charge density defined as qsp

1 = S



ρp dV

(C/m2 )

(3.150)

V

Equation (3.149) can be used to evaluate surface polarization charge density at an interface between two dielectrics. In terms of the total charge density, one can write ǫo ∇ · E = ρ

(3.151)

 ∂ψ2 ∂ψ1 + = qs ∂n1 ∂n2

(3.152)

leading to −ǫo



where qs is the total charge density at an interface.

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ELECTROSTATICS

In this section, the relevant boundary conditions were derived together with expressions for the free, polarization, and total charge densities at an interface. When a body is subjected to an electric field, stresses develop within the body as well as at its outer boundary. When the body is elastic, the electric stresses can deform it. In the next section, we will develop expressions for the electrostatic stresses caused by an electric field.

3.6 MAXWELL STRESS FOR A LINEAR DIELECTRIC It was pointed out that Poisson’s equation, Eq. (3.116), describes the electric potential in a dielectric subject to boundary conditions associated with a given physical situation. In the case of an interface, i.e., a boundary separating two media having different permittivities, the appropriate boundary conditions were discussed. It remains now to evaluate the force exerted on a dielectric body in an electric field. The Korteweig–Helmholtz electric force per unit volume, f, for an incompressible fluid is given as 1 (3.153) f = ρf E − E 2 ∇ǫ 2 This expression is obtained by neglecting a third contribution to the volumetric force density called electrostriction. The electrostriction term is given by ∇



1 2 ∂ǫ E ρ 2 ∂ρ



where ρ is the fluid material density. For further details on the formulation of the electrical body force, see Saville (1997). In all problems discussed in the present book, the strictional term can be omitted. The term ρf E in Eq. (3.153) represents the body force due to the interaction of the free charges in the fluid with the electric field. The last term in Eq. (3.153) accounts for the inhomogeneity in the permittivity of the medium. Recognizing that for a linear dielectric material D = ǫE, ∇ · D = ρf , and after some vector manipulation, one can show that the electric force per unit volume is given by  = 1 f = ∇ · ǫEE − ǫE · EI 2 

(3.154)

The term within the square brackets in Eq. (3.154) is tensorial, and is generally referred = to as the Maxwell stress tensor, T , in a rest frame. Thus, the force per unit volume in a given dielectric is =

where

f =∇ ·T

(3.155)

= = 1 T = ǫEE − ǫE · EI 2

(3.156)

3.6 MAXWELL STRESS FOR A LINEAR DIELECTRIC

69

=

In Eqs. (3.154) and (3.156), the second order unit tensor, I , is defined as 1 I = 0 0 =

0 1 0



 0 0  1

(3.157)

and the term EE is a dyadic product, i.e., a second order tensor, defined as E12  EE = E2 E1 E3 E1 

 E1 E3 E2 E3  E32

E1 E2 E22 E3 E2

(3.158)

where E1 , E2 , and E3 are the components of the vector E that represents the field strength. The force arising from Maxwell electric stress on a body of volume V enclosed by surface S is given by F=



f dV =



=

(∇ · T )dV

(3.159)

V

V

Utilizing the divergence theorem, Eq. (3.159) gives F=



=

(∇ · T )dV =



=

(n · T )dS

S

V =

Recognizing that the Maxwell stress tensor T is symmetric,1 the force on the body due to Maxwell stress can be written as  = (3.160) F = (T · n)dS S

where n is the unit outward normal vector to the surface S enclosing the volume V , as shown in Figure 3.18. From a physical point of view, one can view the stress tensor as that quantity which when dotted with the outward normal n, gives the stress vector acting on a surface whose outward normal is n (Whitaker, 1968). Using Eqs. (3.156) and (3.160), the force acting on a body due to Maxwell stresses becomes    = 1 (3.161) ǫEE − ǫE · EI · n dS F= 2 S Once the electric field strength, E, is known, the force on a body in an electric field can be evaluated through the use of Eq. (3.161). The force acting on the body is

1

=

=

=

For the general case, T · n = n · T , unless the tensor T is symmetric.

70

ELECTROSTATICS

Figure 3.18. A body of volume V bounded by a surface S with an outward normal unit vector n.

to be computed using the unit outward normal vector of the body surface under consideration. If one is interested in evaluating the force acting at a surface element straddling an interface as shown in Figure 3.17, the force is still to be evaluated using the unit outward normal vector. In this respect, the stress tensor at the interface would arise from both sides of the interface. For the case of stress tensor discontinuity, Eq. (3.161) still holds. The implication will be illustrated through several examples. = One can expand the expression for Maxwell stress tensor, T , given by Eq. (3.156) to obtain the components of the stress tensor in Cartesian coordinates.   2 1 2   Ex − 2 E Ex Ey Ex Ez Txx Txy Txz = Ey Ez  Ey2 − 21 E 2 T = Tyx Tyy Tyz  = ǫ  Ey Ex (3.162) Tzx Tzy Tzz Ez Ex Ez Ey Ez2 − 21 E 2 Here, the stress component Tij represents the stress on the i th plane in j -direction resulting from the electric field. For example, Tyx is the stress acting on the y- surface in x-direction. One should note that the stress tensor is symmetric, i.e., Tij = Tj i . The subscript for E signifies the component of the E vector and one can write E 2 = Ex2 + Ey2 + Ez2

(3.163)

As the electric field strength E is related to the electric potential by E = −∇ψ

(3.164)

the explicit relationships in the Cartesian coordinate system become ∂ψ ∂x ∂ψ Ey = − ∂y ∂ψ Ez = − ∂z Ex = −

(3.165-a) (3.165-b) (3.165-c)

3.6 MAXWELL STRESS FOR A LINEAR DIELECTRIC

71

We note that all the components of the field vector E can be related to the scalar potential ψ. The stress tensor can be readily written for other coordinate systems. For example, in spherical coordinates, one can write 

Trr Tθ r Tφr

Trθ Tθ θ Tφθ

  2 1 2 Er − 2 E Trφ Tθ φ  = ǫ  Eθ Er Tφφ Eφ Er

Er Eθ Eθ2 − 21 E 2 Eφ Eθ

 Er Eφ Eθ Eφ  Eφ2 − 21 E 2

(3.166)

where Er , Eθ , and Eφ are the components of E in r, θ, and φ directions, respectively. In spherical coordinates, Eq. (3.164) provides ∂ψ ∂r 1 ∂ψ Eθ = − r ∂θ 1 ∂ψ Er = − r sin θ ∂φ

(3.167-a)

Er = −

(3.167-b) (3.167-c)

Here, E 2 is given by E 2 = Er2 + Eθ2 + Eφ2

(3.168) =

Let us now discuss the implication of the term (T · n) given by Eq. (3.160). In = Cartesian coordinates, the dot product of the stress tensor, T , with the unit normal vector n is given by =

T · n = (Txx nx + Txy ny + Txz nz )ix + (Tyx nx + Tyy ny + Tyz nz )iy

(3.169)

+ (Tzx nx + Tzy ny + Tzz nz )iz where the outward normal vector is given by n = nx ix + ny iy + nz iz

(3.170)

If we have a planar surface whose normal is aligned along the unit vector ix , i.e., n = nx ix , then Eq. (3.169) can be written as =

T · n = Txx nx ix + Tyx nx iy + Tzx nx iz

(3.171)

Although for the unit normal vector n, the quantity nx will be unity, we have retained it explicitly in Eq. (3.171) to ensure that the direction of the unit outward vector is clearly indicated in the analysis. To illustrate the procedure of obtaining the net force on an interface between two dielectrics where discontinuity exists, let us consider a volume element of infinitesimal

72

ELECTROSTATICS

Figure 3.19. Volume element of infinitesimal thickness straddling an interface. The normal to the interface is directed along the x axis.

thickness straddling a surface normal to the x-axis as shown in Figure 3.19. The surface represents the interface between two dielectrics. Accounting for the dielectric media on both sides of the volume element, the net stress acting on the interface is given by =

(T · n)net = [(Txx nx )1 + (Txx nx )2 ] ix   + (Tyx nx )1 + (Tyx nx )2 iy   + (Tzx nx )1 + (Tzx nx )2 iz

(3.172)

Here, the subscripts 1 and 2 denote the media 1 and 2, respectively. One can think = of (T · n) as being the local force per unit area. Consequently, one can write more specifically for the force acting on the entire surface (denoted by S) as F=



=

(T · n)net dS

(3.173)

S

or, F=

    = 1 ǫEE − ǫE 2 I · n dS 2 S net

(3.174)

It should be emphasized that the Maxwell stress tensor due to the electric field as defined above is that of the surrounding medium acting on the surface element whose normal is directed “into” the medium.

3.7 MAXWELL’S EQUATIONS OF ELECTROMAGNETISM

73

In the next chapter, we shall utilize the electrostatic equations, boundary conditions, and Maxwell stress to evaluate the electric field strength, electric potential, surface charge density, and force acting on bodies placed in an external electric field.

3.7 MAXWELL’S EQUATIONS OF ELECTROMAGNETISM In the previous section we dealt with Maxwell’s equations as related to electrostatics. For completeness, the generalized forms of Maxwell’s equations for electromagnetism will be briefly presented in this section. Maxwell’s equations are not limited to electrostatics, but in their general form, describe the laws of electromagnetism. These equations relate the electric field E and the magnetic field H to material properties, the charge density, and conservation principles. The full set of Maxwell’s equations in rationalized MKS (RMKS) units is ∇ · D = ρf

(3.175)

∇ ·B=0

(3.176)

∇ ×E=−

∂B ∂t

(3.177)

and ∇ ×H=

∂D + Jf ∂t

(3.178)

Here, D is the electric displacement vector, ρ is the charge density, B is the magnetic induction field (Tesla, T = NA−1 m−1 ), E is the electric field, H is the magnetic field (Am−1 ), and Jf is the conduction (or free) current density (C/m−2 s−1 ) associated with the movement of free charge. For a linear and homogeneous medium, the following constitutive relationships can be written D = ǫE

(3.179)

B = µH

(3.180)

and where ǫ is the dielectric constant and µ is the magnetic permeability of the material. Utilizing Eqs. (3.179) and (3.180), the Maxwell equations can be written for a linear homogeneous medium as ρf ∇ ·E= (3.181) ǫ ∇ ·H=0 ∇ × E = −µ

(3.182) ∂H ∂t

(3.183)

74

ELECTROSTATICS

and ∂E + Jf (3.184) ∂t It is clear from the above equations that when the electric field varies with time, one needs to consider the magnetic field as well. In this case, the equations governing the electric and magnetic fields are coupled. However, in the absence of time variation, the electric field equations, i.e., electrostatics, can be decoupled from the magnetic equations or magnetostatics. Another point worth noting about the above set of equations is that although these are fundamental relationships governing propagation of electromagnetic waves (i.e., light), the speed of light does not appear in these equations explicitly. This important parameter is, in fact, immediately recovered by noting that ∇ ×H=ǫ

ǫo µo =

1 c2

(3.185)

where ǫo = 8.854 × 10−12 C2 /N.m2 and µo = 1.2566 × 10−7 Henry/m are the permittivity and magnetic permeability of vacuum, respectively, and c is the speed of light in vacuum (2.9979 × 108 m/s). For other material media, the speed of light evolves from the dielectric constant (ǫ) and magnetic permeability (µ) in a manner analogous to Eq. (3.185).

3.8 NOMENCLATURE A B C c D d E E F F12 f H = I i i12 Jf K l N n

cross sectional area of parallel plate capacitor plates, m2 magnetic induction field, Tesla capacitance, Farad speed of light, m/s electric displacement vector, C/m2 vector orientation between two point charges electric field strength, V/m magnitude of electric field, V/m force, N force exerted by charge Q1 on charge Q2 force per unit volume, N/m3 magnetic field, A/m unit tensor unit vector unit vector in the direction of the separation vector R12 conduction or free current density, C/m2 .s conductance, Siemens length, m number of dipoles per unit volume outward unit surface normal vector

3.9 REFERENCES

P Q Qf Qi Qs qs qsf qsp R R12 r r S = T Tij t V W12

75

polarization field strength, C/m2 total electric charge or charge at a point, C total electric free charge on a body, C i th point charge, C surface charge, C surface electric charge density, C/m2 free surface electric charge density, C/m2 polarization surface charge density, C/m2 distance between two points in space, m Resistance, Ohm separation or displacement vector, m position vector, m radial position, m surface area, m2 Maxwell stress tensor, N/m2 component of stress tensor (stress on i th face in j -direction), N/m2 unit tangent vector along a line or plane volume, m3 work done to move a charge along a path 1 to 2, J

Greek Symbols V ǫ ǫo ǫr χ λ µ µo ψ ψs ρ ρf ρp ∇ ∇2

Electric potential difference, V dielectric permittivity of medium, C2 N−1 m−2 or CV−1 m−1 permittivity of vacuum (free space), C2 N−1 m−2 or CV−1 m−1 relative dielectric constant, dimensionless electric susceptibility of the dielectric medium, dimensionless line charge density, C/m magnetic permeability of material medium, Henry/m magnetic permeability of vacuum, Henry/m electric potential, V surface electric potential, V total electric volume charge density, C/m3 material density, kg/m3 free electric charge density, C/m3 polarization charge density, C/m3 del operator, m−1 Laplacian operator, m−2

3.9 REFERENCES Archer, D. G., and Wang, P., The dielectric constant of water and Debye-Hückel limiting law slopes, J. Phys. Chem. Ref. Data, 19, 371, (1990). Eyges, L., The Classical Electromagnetic Field, Dover, New York, (1980).

76

ELECTROSTATICS

Feynman, R. P., Leighton, R. B., and Sands, M., The Feynman Lectures on Physics, vol. II, Addison-Wesley, Reading, MA, (1964). Griffiths, D. J., Introduction to Electrodynamics, Prentice-Hall, Upper Saddle River, NJ, (1989). Israelachvili, J. N., Intermolecular and Surface Forces, Academic Press, London, (1985). Saville, D. A., Electrohydrodynamics: The Taylor-Melcher leaky dielectric model, Ann. Rev. Fluid Mech., 29, 27–64, (1997). Slater, J. C., and Frank, N. H., Electromagnetism, Dover, New York, (1969). Whitaker, S., Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, NJ, (1968).

CHAPTER 4

APPLICATION OF ELECTROSTATICS

In this chapter, we will present different physical problems in the area of electrostatics and seek solutions for the electric field, Maxwell electrostatic stress, and surface charge. It is hoped that through the solution of these problems we will elucidate the concept of electrostatics and provide a foundation for the study of the electrical double layer encountered in the area of electrokinetics. The concept of the electrical double layer will be discussed in Chapter 5. The problems discussed in this chapter are entirely within the scope of the principles outlined in Chapter 3, and hence, may be construed as additional examples for the previous chapter. However, these examples underscore some key electrostatic phenomena that are generally neglected in discussions of electrical double layer interactions. In particular, the scenarios discussed here are applicable to dielectric media without any free mobile charges, i.e., free ions. Such systems are extremely important in a variety of situations.

4.1 TWO-DIMENSIONAL DIELECTRIC SLAB IN AN EXTERNAL ELECTRIC FIELD Consider a flat polymer slab of permittivity ǫ2 and thickness 2d located in a dielectric medium of permittivity ǫ1 . The potential is held at ψA at x = −X and ψB at x = X. The geometry is shown in Figure 4.1. We will evaluate the electric field and potential for the system together with surface polarization charge and the Maxwell electric stresses. Electrokinetic and Colloid Transport Phenomena, by Jacob H. Masliyah and Subir Bhattacharjee Copyright © 2006 John Wiley & Sons, Inc.

77

78

APPLICATION OF ELECTROSTATICS

Region 2 y

Region 3

Region 1

x

Surface R

Surface L at x = -X

2d

at x = X

Figure 4.1. Polymer slab in a dielectric medium with an imposed electric potential.

4.1.1

Electric Potential and Field Strength

In the absence of free charge in the polymer slab and the intervening medium, the governing equation is given by the Laplace equation, Eq. (3.117). For a one-dimensional geometry, one can write Laplace’s equation in one-dimension (along the x-coordinate) for the regions flanking the slab on the left and right sides and for the slab itself: ∂ 2 ψj =0 ∂x 2

(4.1)

where j signifies the region of application. The solution of Eq. (4.1) for the three regions is given by ψ1 = a1 + b1 x

for −d ≥ x ≥ −X

(4.2)

ψ2 = a2 + b2 x

for d ≥ x ≥ −d

(4.3)

ψ3 = a3 + b3 x

for X ≥ x ≥ d

(4.4)

The applicable boundary conditions are:

ǫ1

ǫ2

dψ1 dx dψ2 dx

ψ1 = ψA

at x = −X (Imposed potential)

(4.5-a)

ψ1 = ψ2 dψ2 = ǫ2 dx ψ2 = ψ3 dψ3 = ǫ1 dx ψ3 = ψB

at x = −d (Continuity of potential)

(4.5-b)

at x = −d (No surface free charge)

(4.5-c)

at x = d (Continuity of potential)

(4.5-d)

at x = d (No surface free charge)

(4.5-e)

at x = X (Imposed potential)

(4.5-f)

4.1 TWO-DIMENSIONAL DIELECTRIC SLAB IN AN EXTERNAL ELECTRIC FIELD

79

The boundary conditions provide six equations which are used to evaluate the six constants in Eqs. (4.2)–(4.4). Applying the above boundary conditions leads to (ψA + ψB ) − Cd(ǫ2 − ǫ1 )(ψA − ψB ) − Cǫ2 (ψA − ψB )x 2 (ψA + ψB ) ψ2 = − Cǫ1 (ψA − ψB )x 2 (ψA + ψB ) + Cd(ǫ2 − ǫ1 )(ψA − ψB ) − Cǫ2 (ψA − ψB )x ψ3 = 2 ψ1 =

where C=

1 2[ǫ2 X − d(ǫ2 − ǫ1 )]

(4.6) (4.7) (4.8)

(4.9)

For the special cases of X≫d X=d

1 2ǫ2 X 1 C= 2ǫ1 d

C=

(4.10-a) (4.10-b)

The electric potential on the left of the slab, the slab itself, and on the right of the slab are given by Eqs. (4.6)–(4.8), respectively. In the absence of the slab, one can write ǫ1 = ǫ2 , for which case, Eqs. (4.6)–(4.8) become identical, and are given by ψ=

ψA + ψB ψ A − ψB − x 2 2X

(4.11)

This represents a linear variation of ψ with x. At the origin, the potential is the average of ψA and ψB and the slope of ψ vs. x is −(ψA − ψB )/2X, which is the potential gradient. The electric field strength in the x-direction is given by Eq. (3.65) as Ex = −

dψ dx

(4.12)

Application of Eq. (4.12) to the three regions leads to E1x = Cǫ2 (ψA − ψB )

(4.13)

E2x = Cǫ1 (ψA − ψB )

(4.14)

E3x = Cǫ2 (ψA − ψB )

(4.15)

Equations (4.13)–(4.15) show that the field strength is constant within each of the three regions. The above equations indicate that when the slab permittivity ǫ2 is larger than

80

APPLICATION OF ELECTROSTATICS

the surrounding medium permittivity ǫ1 , i.e., for ǫ2 > ǫ1 , the field strength within the slab is reduced as compared to the field strength in the outside medium. For the case ǫ2 < ǫ1 , the field strength within the slab is higher than that in the outside medium. This is simply a consequence of the polarization charge at the slab’s surfaces. The evaluation of the surface charge density follows in the next section. 4.1.2

Polarization Surface Charge Density

The polarization surface charge density, qsp , for a dielectric is given by Eq. (3.149). Consider the two surfaces L and R in Figure 4.2. For surface L, n1 = −ix and n2 = ix . Application of Eq. (3.149) on surface L leads to (qsp )L = −(ǫ1 − ǫo )

dψ1 dψ2 + (ǫ2 − ǫo ) dx dx

(4.16)

Substituting for the derivative of the potentials, the surface charge density due to polarization for surface L becomes (4.17)

(qsp )L = −Cǫo (ǫ2 − ǫ1 )(ψA − ψB ) In a similar manner, applying Eq. (3.149) to the surface R, one obtains

(4.18)

(qsp )R = Cǫo (ǫ2 − ǫ1 )(ψA − ψB )

As would be expected, the polarization surface charge density on the left and right surfaces are equal in their magnitudes but opposite in sign. When the outer medium is free space, the permittivity ǫ1 becomes equal to ǫo . Recognizing that the constant C and (ǫ2 − ǫo ) are positive quantities, Eq. (4.17) y n2

D

C

A c

B

ix

n1

Surface L

h

b

–ix

n2 d

n1

a

g

–ix

ix

n2

x

n1 i

f

Surface R

Figure 4.2. Surface elements for electrostatic force evaluation for a polymer slab in an external electric field.

4.1 TWO-DIMENSIONAL DIELECTRIC SLAB IN AN EXTERNAL ELECTRIC FIELD

E1x

− − − − − −

E2x

+ + + + + +

E1x

(a)

+ + + + + +

E2x

81

– – – – – –

(b)

Figure 4.3. Polarization surface charge density for two dielectrics. The electric field inside the slab will be reduced for (a) ǫ2 > ǫ1 and enhanced for (b) ǫ2 < ǫ1 .

would indicate that when the polymer slab is in free space, a negative charge density will always develop on the left hand side of the slab. The situation is different when the slab is in another dielectric medium. The polarization charge density on the left hand side would depend on the sign of (ǫ2 − ǫ1 ). When ǫ2 > ǫ1 , the left hand side surface will carry a negative charge, while for ǫ2 < ǫ1 , it will carry a positive charge. Consequently, the electric field strength as indicated by Eqs. (4.13) and (4.14) in the slab will be E2x < E1x

for

ǫ2 > ǫ1

(4.19-a)

E2x > E1x

for

ǫ2 < ǫ1

(4.19-b)

E2x = E1x

for

ǫ2 = ǫ1

(4.19-c)

as shown in Figure 4.3. 4.1.3

Maxwell Electrostatic Stress

In this section, we will derive expressions for the Maxwell stresses for the left and right hand side surfaces. As the physical problem is one-dimensional, the analysis will be relatively straightforward since we only have to deal with the electric field component Ex and the electrostatic stress component Txx . Here, the stress component Txx is by definition the stress exerted by the outer environment on the x-plane in the x-direction. In all our analysis we assume that the medium permittivity is a constant. From Eq. (3.162), the Maxwell stress, Txx , is given by Txx = ǫ




Ex2

E2 − 2



(4.20)

As we are dealing with a one-dimensional problem where Ey = Ez = 0, one can write E = Ex

(4.21)

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APPLICATION OF ELECTROSTATICS

and Eq. (4.20) becomes ǫEx2 (4.22) 2 In order to evaluate the net force per unit area on a surface, it is necessary to perform a surface integration given by Eq. (3.160), which requires the Maxwell electrostatic stress components. Figure 4.2 shows the integration paths for the evaluation of the electrostatic forces. Applying Eqs. (3.172) and (3.173) for the surface element abcd located on surface L, the x-directed force per unit area, FxL , is given by Txx =

FxL ix = (n1 Txx )1 + (n2 Txx )2

(4.23)

Recognizing that n1 = −ix and n2 = ix , Eq. (4.23) becomes FxL = −(Txx )1 + (Txx )2

(4.24)

Here, (Txx )1 signifies the evaluation of the stress component Txx in medium 1 (surface cd) and (Txx )2 denotes evaluation of Txx in medium 2 (surface ab). Making use of Eq. (4.22), one can write for Eq. (4.24) 1 1 2 2 + ǫ2 E2x FxL = − ǫ1 E1x 2 2

(4.25)

with the use of expressions for the field strength provided by Eqs. (4.13) and (4.14), Eq. (4.25) becomes FxL = −

C2 C2 ǫ1 ǫ22 (ψA − ψB )2 + ǫ2 ǫ12 (ψA − ψB )2 2 2

Upon simplification we obtain FxL = −

C2 ǫ1 ǫ2 (ǫ2 − ǫ1 )(ψA − ψB )2 2

(4.26)

Clearly, as C 2 ǫ1 ǫ2 (ψA − ψB )2 /2 is always positive, the direction of the force on surface L is dependent on the sign of (ǫ2 − ǫ1 ). Let us now evaluate the x-directed force per unit area, FxR , on the surface R. Applying Eqs. (3.172) and (3.173) on surface element f ghi, we have FxR ix = (nTxx )1 + (nTxx )2

(4.27)

where the unit outward normal vectors n1 and n2 for surface R are given by n1 = ix

and

n2 = −ix

(4.28)

The x-directed force per unit area on surface R becomes FxR = (Txx )1 − (Txx )2

(4.29)

83

4.2 A DIELECTRIC SPHERE IN AN EXTERNALELECTRIC FIELD

From Eqs. (4.22), (4.14) and (4.15), we write for Eq. (4.29) FxR =

C2 ǫ1 ǫ2 (ǫ2 − ǫ1 )(ψA − ψB )2 2

(4.30)

Comparing the electrostatic forces on surfaces L and R shows that FxL + FxR = 0. In other words, the net force on the polymer slab is zero. This result should not be surprising as the total charge of the slab is zero. A linear dielectric body with zero total charge does not experience a force arising from its placement in a uniform electric field strength. It should be noted that with the absence of free charge, the polarization charge represents the total charge, cf., Eq. (3.101). As the net force on the slab is zero, the polymer plate would not move in any preferred direction in the external electric field and will remain stationary. A more straightforward approach to evaluate the electric force on the slab would be to use the force acting on the element ABCD as shown in Figure 4.2. Although the net force on the polymer slab is zero, the forces acting on surfaces L and R can either be compressive or tensile. When ǫ2 > ǫ1 the slab is in tension and when ǫ2 < ǫ1 , the slab is in compression. Consequently, if the polymer is made of an elastic dielectric material, then under the external electric field, the slab’s thickness increases for ǫ2 > ǫ1 and decreases for ǫ2 < ǫ1 .

4.2 A DIELECTRIC SPHERE IN AN EXTERNAL ELECTRIC FIELD Consider a spherical drop of a dielectric fluid (e.g., an oil) of a constant permittivity ǫ2 being placed in another dielectric fluid of constant permittivity ǫ1 . The initial uniform electric field strength is Eo and it is parallel to the z-direction. Assume that the oil drop has no electric charge prior to its placement in the electric field. The physical configuration is shown in Figure 4.4. It is of interest to evaluate the electric field and potential, together with the polarization charge and the Maxwell electric stress for the oil drop.

Figure 4.4. A dielectric sphere in another dielectric under an external electric field.

84

APPLICATION OF ELECTROSTATICS

4.2.1

Electric Potential and Field Strength

Since there is no free charge inside and outside the sphere, Laplace’s equation is applicable and it is given by ∇ 2 ψi = 0

(4.31)

for i = 1, 2 representing outside and inside the sphere, respectively. A spherical coordinate system is best suited for analyzing the problem, and we can write the Laplace equation for outside and inside the sphere as Outside the sphere:

 1 ∂ ∂ 1 ∂ψ1 2 ∂ψ1 r + sin θ =0 (4.32) r 2 ∂r ∂r r 2 sin θ ∂θ ∂θ Inside the sphere: 1 ∂ r 2 ∂r




r2

∂ψ2 ∂r



+

∂ 1 2 r sin θ ∂θ




sin θ

∂ψ2 ∂θ



=0

(4.33)

The boundary conditions are ψ2 ψ1 = ψ2 ǫ1

∂ψ1 ∂ψ2 = ǫ2 ∂r ∂r

is finite at r = 0

continuity of potential at r = a

(4.34) (4.35)

no surface free charge (continuity of displacement) at r = a (4.36) E = iz Ez = iz Eo at r → ∞ (4.37)

The last boundary condition states that the electric field is undisturbed at locations far from the sphere and it is in z-direction. With two second order differential equations, the four boundary conditions, Eqs. (4.34)–(4.37), define the physical problem. Before we proceed with the solution, let us examine the boundary condition far from the sphere. The uniform electric field, Eo , at r → ∞ can be decomposed into Er and Eθ components as shown in Figure 4.4. Er = Eo cos θ

(4.38)

Eθ = −Eo sin θ

(4.39)

The components of the electric field strength in spherical coordinates are given by ∂ψ1 ∂r 1 ∂ψ1 Eθ = − r ∂θ Er = −

(4.40-a) (4.40-b)

4.2 A DIELECTRIC SPHERE IN AN EXTERNALELECTRIC FIELD

85

Combining Eqs. (4.38)–(4.40-b) leads to ∂ψ = −Eo cos θ ∂r ∂ψ = Eo r sin θ ∂θ

(4.41-a) (4.41-b)

Solutions of Eqs. (4.40) and (4.41) are given by, respectively, ψ1 = −Eo r cos θ + g(θ)

(4.42)

ψ1 = −Eo r cos θ + f (r)

(4.43)

and Comparison between Eqs. (4.42) and (4.43) indicates that g(θ) = f (r) = C

(4.44)

As we are normally interested in an electrical potential difference, we arbitrarily set the constant C to zero, yielding ψ1 = −Eo r cos θ

(4.45)

Equation (4.45) gives the boundary condition in terms of the electric potential for an undisturbed electric field. The boundary condition is applicable at r → ∞ and it is more convenient to use in our analysis. This boundary condition suggests that the electric potential should vary in θ-direction as the cosine of the angular position θ. Consequently, a possible solution (see Section 4.3) is: ψ1 =




 A1 + A r cos θ 2 r2

(4.46)

ψ2 =




 B1 + B r cos θ 2 r2

(4.47)

and

The constants A1 , A2 , B1 , and B2 can now be evaluated from the boundary conditions. The solution is given as Outside the sphere: 
  G − 1 a3 ψ1 = E0 −r + cos θ G + 2 r2

(4.48)

and, inside the sphere: ψ2 = −

3Eo r cos θ G+2

(4.49)

86

APPLICATION OF ELECTROSTATICS

where G=

ǫ2 ǫ1

(4.50)

Recognizing that


E = − ir

1 ∂ψ ∂ψ + iθ ∂r r ∂θ



(4.51)

the electric fields are given by Outside the sphere:    
 G − 1 a3 2(G − 1) a 3 E1 = ir Eo 1 + cos θ + iθ Eo −1 + sin θ (G + 2) r 3 G + 2 r3

(4.52)

and, inside the sphere: E2 = ir

3Eo cos θ 3Eo sin θ − iθ G+2 G+2

(4.53)

Close examination of Eq. (4.53) yields 3Eo iz (G + 2)

E2 =

(4.54)

Equation (4.54) states that the electric field inside the sphere is uniform and it is in z-direction. Its value relative to the external electric field is dependent on G which is given by ǫ2 /ǫ1 . The inner electric field strength is smaller than Eo when the sphere’s permittivity is larger than that on the outside, i.e., G > 1. Similar conclusions were obtained for the case of the slab in the previous example. 4.2.2

Polarization Surface Charge Density

Evaluation of the polarization surface charge density is a direct application of Eq. (3.149), which is given as qsp = (ǫ1 − ǫo )

∂ψ1 ∂ψ2 + (ǫ2 − ǫo ) ∂n1 ∂n2

(4.55)

As shown in Figure 4.5, the unit vectors n1 and n2 are related to ir leading to n1 = r

and n2 = −r

(4.56)

Making use of the expressions for ψ1 and ψ2 , and Eq. (4.56), Eq. (4.55) gives the local surface polarization charge density as qsp = 3ǫo Eo




 G−1 cos θ G+2

(4.57)

4.2 A DIELECTRIC SPHERE IN AN EXTERNALELECTRIC FIELD

87

Figure 4.5. A dielectric sphere in an external electric field showing outward normal vectors, a surface element, and the Maxwell stresses.

The distribution of the polarization charge varies as the cosine of the angle θ. Furthermore, its sign is dependent on whether G is greater or smaller than unity. The total surface polarization charge is given by Qsp = 3ǫo Eo




G−1 G+2



π

2π a 2 sin θ cos θ dθ

(4.58)

0

The surface element for the integration is shown in Figure 4.5. Integration of Eq. (4.58) gives Qsp = 0. As a whole, the oil drop remains electro-neutral after its placement in the electric field and it is in agreement with the first fundamental principle of electrostatics. 4.2.3

Maxwell Electrostatic Stress on the Dielectric Sphere

When the dielectric sphere is placed in the electric field, surface stresses develop due to the electric field. The local force can be evaluated from Eq. (3.173), where  = F= (T · n)net dS (4.59) surface

By analogy to Eq. (3.169), one can write =

T · n = (Trr nr + Trθ nθ )ir + (Tθ r nr + Tθ θ nθ )iθ

(4.60)

where nr and nθ are the components of the outer unit vector n. Here, ir and iθ are unit vectors along the r and θ coordinates, respectively. The outer spherical surface at r = a is of interest to us and it is at this surface that we need to evaluate the Maxwell stresses. For this surface, Eq. (4.60) reduces to =

T · n = Trr nr ir + Trθ nr iθ

(4.61)

88

APPLICATION OF ELECTROSTATICS

By analogy to Eq. (3.172) one can write the local net force per unit area acting on the outer surface (r = a for all θ) as =

(T · n)net = [(Trr )1 − (Trr )2 ] ir + [(Trθ )1 − (Trθ )2 ] iθ

(4.62)

=

The term (T · n)net represents the local net force per unit area acting on the surface. The components of the Maxwell stress tensor are provided by Eq. (3.166) Trr = ǫ




Er2

E2 − 2



(4.63)

and (4.64)

Trθ = ǫEθ Er

In Eqs. (4.63) and (4.64), the permittivity and electric field are evaluated for the medium under consideration. Let us now evaluate the terms in Eq. (4.62). Consider first the term (4.65)

Fr = [(Trr )1 − (Trr )2 ]

where Fr represents the local net force per unit area in r-direction. From Eq. (4.63), we can substitute for the electric field components in Eq. (4.65) to give 

  E2 E2 2 2 − ǫ Er − Fr = ǫ Er − 2 2 1 2

(4.66)

Let us now consider the term  
E2 ǫ Er2 − 2 1 Setting r = a and making use of Eq. (4.52), we can write E12

=

Eo2



2(G − 1) 1+ (G + 2)

2

2

cos θ +

Eo2



−1 +




G−1 G+2

2

sin2 θ

(4.67)

leading to E12 = Now

leading to




3Eo G+2

2



G2 cos2 θ + sin2 θ

  2(G − 1) 2 2 cos2 θ E1r = Eo2 1 + (G + 2) 2 E1r

=




3GEo G+2

2

cos2 θ



(4.68)

(4.69)

(4.70)

4.2 A DIELECTRIC SPHERE IN AN EXTERNALELECTRIC FIELD

89

Consequently we can write for the first term of Eq. (4.66) 

  3Eo 2  2 ǫ1 E2 = ǫ Er2 − G cos2 θ − sin2 θ 2 2 G+2 1

(4.71)

Now let us consider the term


 E2 ǫ Er2 − 2 2 From Eq. (4.53), we can write E22 =




3Eo cos θ G+2

2

+




3Eo sin θ G+2

2

(4.72)

leading to E22

=

consequently we can write

2

=




 3Eo cos2 θ G+2

Now, 2 E2r

3Eo G+2






 E2 3Eo 2 ǫ2 ǫ Er2 − [2 cos2 θ − 1] = 2 2 G + 2 2

(4.73)

(4.74)

(4.75)

Combining Eqs. (4.71) and (4.75) and rearranging, the local net radial force per unit area, Fr , exerted by the surrounding medium on the dielectric oil drop is given by Fr =

ǫ1 2




3Eo G+2

2

[(G − 1) + (G − 1)2 cos2 θ ]

(4.76)

Evaluation of ǫEθ Er as given in Eq. (4.64) for the two media leads to (Trθ )1 = (Trθ )2

(4.77)

This would indicate that there is no net force in the angular direction. In other words, the oil drop does not experience a shearing force due to the electric field. It is rather interesting to obtain a physical meaning for the net local radial force given by Eq. (4.76). The expression has two terms. The first term can be either positive or negative depending on the relative magnitude of the permittivities of the two dielectrics. Irrespective of the sign of (G − 1), the first term implies a constant radial force on the spherical oil drop and it does not contribute to the drop’s deformation. The second term is always positive and it implies that the radial force is dependent on

90

APPLICATION OF ELECTROSTATICS

angle θ. As cos2 θ attains a maximum value at θ = 0 and π , while it attains a minimum value at θ = π/2, one can conclude from Eq. (4.76) that the oil droplet will always elongate along the electric field. Water drop deformation in a electric field is shown in Figure 4.6. Drop deformation analysis for perfect dielectrics under the influence of an electric field was conducted by O’Konski and Thatcher (1953) and Allan and Mason (1962). Their theoretical analysis showed that dielectric liquid drops always deformed to a prolate shape (elongation along the direction of the applied field) as suggested by Eq. (4.76) . Observations reported in different publications (O’Konski and Thatcher, 1953; Taylor, 1964; Garton and Krasucki, 1964; Ha and Yang, 2000; Lu, 2002) confirm such a prediction. However, for some systems, many authors noted in their experimental studies that the drops deformed to an oblate shape (Allan and Mason, 1962; Torza et al., 1971; Arp et al., 1980; Vizika and Saville, 1992; Ha and Yang, 1995). In the classical papers by Taylor (1966) and Melcher and Taylor (1969), the drop deformation has been addressed by accounting for the non-zero electric conductivity of both the fluids constituting the droplet and the surrounding medium. These studies were performed by considering the redistribution of the electric field due to the migration of the charge carriers. Such an approach is referred to as the leaky dielectric model. Depending on the relationship between conducting and dielectric properties of the liquids, Taylor’s theory predicts either prolate or oblate shape of the deformed droplet. A generalized analysis of the drop deformation problem was recently proposed by Zholkovskij et al. (2002), who, in addition to the migration transport of the charge carriers, considered their diffusive transport. The final expression of Zholkovskij et al. (2002) provides a smooth bridging between the perfect dielectric (Allan and Mason, 1962) and leaky dielectric (Taylor, 1966) models.

Figure 4.6. Water drop deformation in an electric field of 1.5 kV/cm. The external liquid is 0.5% by volume bitumen obtained from Alberta oil sands in toluene (Lu, 2002).

91

4.3 A CONDUCTING SPHERE IN AN EXTERNAL ELECTRIC FIELD

4.3 A CONDUCTING SPHERE IN AN EXTERNAL ELECTRIC FIELD Consider a sphere made up of a conducting medium being placed in a dielectric medium in an initially uniform electric field, Eo . Assume that the sphere is grounded and prior to its placement in the electric field, it has no free charge. Figure 4.7 shows the geometry of the system. Let us evaluate the system characteristics. 4.3.1

Electric Potential and Field Strength for a Conducting Sphere

As the conducting sphere has no free charge, the Laplace equation can be used to describe the electric potential distribution. It is given by ∇ 2ψ = 0

(4.78)

Here, the electric potential refers to the region outside the conducting sphere. As the electric field strength is zero inside the conducting sphere, it follows that in the inner region ψ is a constant and it is taken to be zero. In spherical coordinates, the Laplace equation has the form

 ∂ 1 ∂ψ 1 ∂ 2 ∂ψ r + sin θ =0 (4.79) r 2 ∂r ∂r r 2 sin θ ∂θ ∂θ The boundary conditions are ψ =0

at

ψ = −Eo r cos θ

r=a

(4.80)

at

(4.81)

r→∞

As we discussed in the previous section, the boundary condition for ψ at r → ∞ signified that the electric field is undisturbed. From Eq. (4.81), the form of the angular variation of ψ is suggested and one can guess the solution for ψ to be of the form ψ = f (r) cos θ

Figure 4.7. A conducting sphere in a uniform electric field Eo .

(4.82)

92

APPLICATION OF ELECTROSTATICS

Substituting Eq. (4.82) in Eq. (4.79) leads to 2r

df df + r2 − 2f = 0 dr dr

By letting f = r n , one can show that the solution of f is given by
 C1 f = + C r 2 r2

(4.83)

The integration constants C1 and C2 can be determined by applying the boundary conditions (4.80) and (4.81), leading to C1 = a 3 Eo

and

C2 = −Eo

where a is the sphere radius. The electric potential outside the sphere becomes 
3 a ψ = Eo − r cos θ (4.84) r2 It is interesting to note that the electric potential does not depend on the medium permittivity. The electric field strength is deduced from Eq. (4.84) where E = Er ir + Eθ iθ = −∇ψ ∂ψ 1 ∂ψ =− ir − iθ ∂r r ∂θ

(4.85)

leading to E = Eo




 
3 2a 3 a + 1 cos θ ir + Eo − 1 sin θ iθ r3 r3

(4.86)

At the sphere surface (r = a), the θ-component for the electric field strength is zero and we obtain from Eq. (4.86) Er |r=a = 3Eo cos θ 4.3.2

Surface Charge Density for a Conducting Sphere

The polarization charge density on the conducting spherical surface is given by Eq. (3.149). With ψ2 = 0 and letting ψ1 = ψ, one can directly write  ∂ψ  (4.87) qsp = (ǫ1 − ǫo ) ∂r r=a From the solution for ψ provided by Eq. (4.84) we can deduce the polarization charge density from Eq. (4.87) to be qsp = −3(ǫ1 − ǫo )Eo cos θ

(4.88)

93

4.3 A CONDUCTING SPHERE IN AN EXTERNAL ELECTRIC FIELD

Eo

− − − − − −

+ + + + + +

Eo x

− − − − − −

(a)

+ + + + + +

x

(b)

Figure 4.8. (a) Surface polarization charge on a conducting sphere. (b) Free surface charge on a conducting sphere.

where ǫ1 is the permittivity of the external material into which the conducting sphere is placed. As (ǫ1 − ǫo ) is always positive, the surface polarization charge distribution does not depend on the medium permittivity. The polarization surface charge is depicted in Figure 4.8(a). As the sphere’s material is a conductor, by placing the sphere in the electric field, unbound electrons would move to the sphere’s surface. Consequently, one would expect that the total surface charge is different from the surface polarization charge. The total surface charge is given by Eq. (3.152). One can write qs = −ǫo

∂ψ ∂ψ1 = −ǫo ∂n1 ∂r

(4.89)

leading to qs = 3ǫo Eo cos θ

(4.90)

Since the difference (qs − qsp ) is termed free surface charge, we can write qsf = 3ǫ1 Eo cos θ

(4.91)

Eq. (4.91) indicates that there is a free surface charge, even though there is no free charge within the inner region of the sphere. The free surface charge is schematically depicted in Figure 4.8(b). The total charge residing on the sphere’s surface is given by qs dS =



= 6π a 2 ǫo Eo



Qs =



S

π

0 π

2π a 2 qs sin θ dθ sin θ cos θ dθ

(4.92)

0

Upon integration, it leads to the expected result Qs = 0

(4.93)

94

4.3.3

APPLICATION OF ELECTROSTATICS

Maxwell Electrostatic Stress on the Conducting Sphere

Let us now evaluate the stress on the surface of the conducting sphere due to its placement in the external electric field. The surface stresses are shown in Figure 4.9. The stress tensor in spherical coordinates is given by
T T = rr Tθ r =

Trθ Tθ θ




2 1 2 Er − 2 E =ǫ Eθ Er

Er Eθ Eθ − 21 E 2



(4.94)

At r = a, one can show that Eθ = 0

(4.95)

E = Er = 3Eo cos θ

(4.96)

and

From Eqs. (4.94) to (4.96) one can write Trθ = 0

(4.97)

9ǫ1 2 E cos2 θ 2 o

(4.98)

and Trr =

For this problem, the inner electric field strength is zero, and hence, the inner Trr stress component is zero.

Trr

Tr n1

Fx n2

r

Eo

x 1

a

Figure 4.9. Surface stresses on a conducting sphere.

95

4.4 CHARGED DISC AND TWO PARALLEL DISCS IN A DIELECTRIC MEDIUM

The net force in x-direction is given by Fx =



0

π

Trr (2π a 2 sin θ ) cos θ dθ

= 9π ǫ1 a 2 Eo2



π

sin θ cos3 θ dθ

(4.99)

0

Upon integration, we obtain Fx = 0. A physical interpretation for this result is that there is no net force on the spherical conductor when it is placed in a uniform electric field. Such a conclusion would be easily reached as the total charge, Qs , is zero and the electric field strength is initially uniform.

4.4 CHARGED DISC AND TWO PARALLEL DISCS IN A DIELECTRIC MEDIUM The purpose of this example is to evaluate the electric field strength due to a charged disc and to eventually evaluate the force between two infinitely large plates carrying a uniform charge density. Let us first discuss the case of the electric field strength at a point A along the z-coordinate, being at a distance z from a flat disc carrying a surface charge density qs1 . The geometry is shown in Figure 4.10. Making use of Eq. (3.16) one can write for the electric field strength at point A E=



S

qs1 i dS 4π ǫR 2

(4.100)

iz z

i

R

A

x x dx L

Figure 4.10. Electric field strength due to a disc carrying a surface charge density, qs1 .

96

APPLICATION OF ELECTROSTATICS

From the geometry, the electric field strength in z-direction is given by Ez =



L

0

2π xqs1 cos θ dx 4π ǫR 2

(4.101)

where L is the disc radius, ǫ is the medium permittivity, i · iz = cos θ, and dS = 2π xdx. Recognizing that R 2 = x 2 + z2

(4.102)

cos θ = z/R

(4.103)

and Equation (4.101) becomes Ez =

zqs1 2ǫ



L 0

xdx (x 2 + z2 )3/2

(4.104)

Making the substitution y = x 2 + z2 and performing the integration in Eq. (4.104) one obtains Ez =

qs1 [(L2 + z2 )1/2 − z] 2ǫ(L2 + z2 )1/2

(4.105)

This is the electric field strength at A. For the case of L ≫ z one can write Ez =

qs1 2ǫ

(4.106)

The above result suggests that the electric field strength is independent of the position of z along the disc axis when the radius of the circular disc is large (L → ∞). At first, this appears to be a surprising result. However, this behavior can be qualitatively explained by noting that when the point A is relatively close to the disc of infinite radius, the major contribution to the field comes from the regions near the origin of the disc, while the peripheral regions of the disc have smaller contributions owing to their large separations from the point A. This is evident from the fact that the field is inversely proportional to R 2 , and that R will be large even for small values of z when x is large, cf., Eq. (4.102). This is analogous to stating that when the point A is close to a disc of infinitely large radius, it “sees” a smaller region of the disc. Now, when the point A moves farther away from the disc in the z-direction, the contribution to the field from the peripheral regions of the disc becomes comparable to that from the central regions – in other words, point A starts “seeing” a larger surface area of the disc. The fact that point A “sees” more of the disc surface as it moves away from the disc leads to a constant electric field strength, see Griffiths (1989) for more details.

4.5 POINT CHARGES IN A DIELECTRIC MEDIUM

97

Let us now evaluate the force between two large parallel circular planes 1 and 2, carrying surface charge densities of qs1 and qs2 , respectively. Let us locate point A on plane 2. Recognizing that a point charge of Q would experience a force of Qqs1 /2ǫ, then by the principle of superposition, the force between the two circular planes is given by Sqs1 qs2 (4.107) F12 = 2ǫ or qs1 qs2 F12 = (4.108) S 2ǫ where S is the surface area of each plate. Equation (4.108) represents the force per unit area between two large plates carrying charge densities of qs1 and qs2 in a dielectric medium with permittivity ǫ. In Chapter 5, we will compare this force per unit area with the situation of two parallel plates in an electrolyte solution where free charge is present.

4.5 POINT CHARGES IN A DIELECTRIC MEDIUM We have discussed in Example 1 of Chapter 3 the electric field strength due to two point charges. Let us extend the analysis to include three point charges. Let us consider three point charges Q1 , Q2 , and Q3 located along the x-axis as shown in Figure 4.11. The consecutive point charges are separated by a distance d/2. The medium has a constant permittivity ǫ. In a dielectric medium, the electric potential due to a point charge at a distance r away from the point charge is given by ψ=

Q 4π ǫr

(4.109)

y A(x,y,z) r3 r2 Q3

Q2 d ⁄2

r1 Q1

x

d ⁄2

z

Figure 4.11. Three point charges equally spaced on x-axis.

98

APPLICATION OF ELECTROSTATICS

By the superposition principle, the electric potential at A due to point charges Q1 , Q2 , and Q3 is given by   1 Q2 Q3 Q1 ψ= (4.110) + + 4π ǫ r1 r2 r3 Rearranging Eq. (4.110) gives 4π ǫrψ =



Q1 Q2 Q3 + + r1 /r r2 /r r3 /r



(4.111)

where r denotes the distance of point A from the origin. From the geometry of Figure 4.11, making use of the cosine law and rearranging, we can write r2 = r
2 d r1 = 1+ − r 2r
2 d r3 = 1+ + r 2r

(4.112-a) d cos θ r

1/2

(4.112-b)

d cos θ r

1/2

(4.112-c)

Substituting for r1 , r2 , and r3 using the above equations, and expanding in Taylor’s series, where n(n − 1) 2 (1 + x)n = 1 + nx + x + ··· 2! with n = −1/2 and x being the second and third terms in Eqs. (4.112-b) and (4.112-c), we can write Eq. (4.111) as 4π ǫrψ 

= Q1 1 −

1 2




d 2r

2

1 + Q2 + Q 3 1 − 2

−




d cos θ r d 2r

2



+

3 8




d + cos θ r



d 2r

2

3 + 8

−




d cos θ r d 2r

2

2



+ ···

d + cos θ r

2



+ ···

(4.113)

Such an expansion is valid when x < 1. Upon rearrangement, Eq. (4.113) yields


2
 Q1 d d 3 cos2 θ − 1 4π ǫψ = cos θ + 1+ + ··· r 2r 2r 2


2
 Q3 d d 3 cos2 θ − 1 Q2 + cos θ + 1− + ··· (4.114) + r r 2r 2r 2

4.5 POINT CHARGES IN A DIELECTRIC MEDIUM

99

Seeking a more compact form for Eq. (4.114), and letting r ≫ d where only two expansion terms are retained, one obtains
 d 1 1 (Q1 + Q2 + Q3 ) + (Q1 − Q3 ) cos θ r 2 r2


2
1 3 cos2 θ − 1 d (4.115) + (Q1 + Q3 ) 2 r3 2

1 ψ(r, θ) = 4π ǫ



The electric potential at point A due to three point charges is given by Eq. (4.115). Let us discuss some of the properties of the electric potential ψ for the case of three point charges. Clearly, the variations of ψ(r, θ) are largely dependent on the signs of the point charges. Let us consider some specific cases: Case (i), Q1 = −Q3 = Q and Q2 = 0 ψ=

Qd cos θ 4π ǫr 2

(4.116)

This is the electric potential due to a dipole. It is in accordance with Eq. (3.34) from Example 1 of Chapter 3, that describes the electric field strength for two point charges of +Q and −Q. In this case the decay of the potential ψ is of the order 1/r 2 . Case (ii), Q1 = Q2 = Q3 = Q Q ψ= 4π ǫ



3 2 + r r




d 2r

2


3 cos2 θ − 1 2



or, for d/r ≪ 1, ψ=

3Q 4π ǫr

(4.117)

In this case with d/r ≪ 1, the potential decays quite slowly as 1/r. One can easily generalize Case  (ii) to the following scenario. Case (iii), Qi = 0 with d/r ≪ 1 ψ= As long as





Qi 4π ǫr

(4.118)

Qi = 0, the decay of the electric potential is proportional to 1/r.

100

APPLICATION OF ELECTROSTATICS

4.6 NOMENCLATURE half-thickness of a dielectric slab, m separation distance between point charges, m electric field strength, V/m magnitude of electric field, V/m magnitude of applied electric field, V/m components of electric field in spherical coordinates, V/m electric field component along x direction, V/m force, N force between two charged planes, N radial force per unit area on a sphere, N/m2 force per unit area along x on surfaces L and R, N/m2 ratio of dielectric permittivities, ǫ2 /ǫ1 unit vector along r direction unit vector along x direction radius of a charged disc, m length, m unit outward surface normal vector total electric charge or charge at a point, C total electric free charge on a body, C i th point charge, C total surface polarization charge, C total surface charge density, C/m2 free surface electric charge density, C/m2 polarization surface charge density, C/m2 position vector, m radial position, m surface area, m2 Maxwell stress tensor, N/m2 components of Maxwell tensor (spherical coordinates), N/m2 component of Maxwell tensor (Cartesian coordinates), N/m2 Cartesian coordinate axes

d E E Eo Er , Eθ Ex F F12 Fr FxL , FxR G ir ix L l n Q Qf Qi Qsp qs qsf qsp r r S = T Trr , Trθ , Tθ θ Txx x, y, z Greek Symbols ǫi ǫo ǫr ψi ψA , ψB ρ θ

dielectric permittivity of medium i, C2 N−1 m−2 permittivity of vacuum (free space), C2 N−1 m−2 relative dielectric constant, dimensionless electric potential in medium i, V surface electric potential, V total electric volume charge density, C/m3 azimuthal coordinate in spherical coordinate system

4.7 PROBLEMS

101

4.7 PROBLEMS 4.1. A cylindrical polymer rod of a dielectric material is placed in another dielectric material and is subjected to an initially uniform electric field Eo as shown in Figure 4.12. The permittivity of the rod is ǫ2 and the permittivity of the outer material is ǫ1 . (a) Show that −2GEo r cos θ for r ≤ a ψ2 = (1 + G) ψ1 = Eo



 a 2 (1 − G) − r cos θ (1 + G)r

for r ≥ a

where G = ǫ1 /ǫ2 . (b) Derive an expression for the polarization charge density on the surface of the rod. (c) Derive an expression for the radial force exerted on the rod. 4.2. Consider once again the flat dielectric slab discussed in Section 4.1. The slab has a permittivity of ǫ2 and it is placed in a dielectric medium of permittivity ǫ1 . The slab has a “glued” surface charge density of qs . In medium 1, where the slab is placed, the potential at x = −X is ψA while the potential at x = X is ψB . The slab thickness is 2d. (a) Discuss the reason why Laplace’s equation is the governing equation for the dielectric potential inside and outside the slab. (b) Write down the applicable boundary conditions for the Laplace equation. (c) Derive the electric potential for the left and right hand sides of the slab, as well as for the slab itself. What are the corresponding field strengths? (d) Derive an expression for the Maxwell electric stress on the slab surfaces.

r

Eo

x a 2

1

Figure 4.12. Cylindrical polymer rod.

102

APPLICATION OF ELECTROSTATICS

(e) What is the net force per unit area exerted on the slab due to the electric field? 4.3. A parallel plate capacitor consists of two metal plates, separated by a distance d, with their surface charges maintained at Q and −Q. The gap between the two surfaces is filled with a dielectric material having a permittivity of ǫ. The capacitance, C, is defined as Q C= V where V is the potential difference between the two plates. (a) Show that for the case of parallel plate capacitor, the capacitance is given by C=

Aǫ d

where A is the area of a plate. (b) For the case of two concentric metal shells, with radii a and b, show that the corresponding capacitance is given by C=

4π ǫab (b − a)

4.4. A metal rod is placed normal to an initially uniform electric field Eo . Derive expressions for (a) Electric potential inside and outside the rod. (b) Electric field strength inside and outside the rod. (c) Induced surface charge. 4.5. We have discussed in Section 4.5 the electric field due to three point charges. We will now attempt to evaluate the electric field potential due to multiple point charges located at different positions. Consider a dielectric medium of a permittivity ǫ in which N point charges Qi are at various locations defined by (ri , θi ) as shown in Figure 4.13. Here Ri = r − ri . Note that in Figure 4.13, the angle θi lies on the plane containing points A, O, and the location of point charge Qi . (a) Show that for point A   r  1    r i 2 1 1 i 3 cos2 θi − 1 + + ··· = 1 + cos θi Ri r r 2 r

Show that for ri /r < 1, the electric field potential is given by

  N  r 2 1  1 ri i 2 + ··· Qi 1 + cos θi + (3 cos θi − 1) ψ(r) = 4π ǫr i=1 r 2 r (4.119)

4.8 REFERENCES

103

A

Ri r

Qi ri

i O

Figure 4.13. Multiple point charges.

(b) The electric potential at a point due to a spherical surface carrying a surface charge density qs was analyzed in Example 3.3 of Chapter 3. Normalize Eq. (3.86) from that example to give ψ(z)ǫo a = aqs z

(4.120)

Now assume that N point charges are uniformly distributed to form a charged spherical shell whose center is at the origin O. Assume that all the point charges have the same charge Q and ri = a for all i. Write a relationship between qs and Q and show that N 1  a  ǫψ(r)  a  cos θi = 1+ aqs r N i=1 r

N  a 2 1 1 2 + + ··· (4.121) (3 cos θi − 1) N i=1 2 r Compare Eqs. (4.120) and (4.121) for N = 100 and r/a = 5 and 100. What is the significance of the limit N → ∞? Comment on your findings. Note that (a/r) is equivalent to (a/z). 4.8 REFERENCES Allan, R. S., and Mason, S. G., Particle behavior in shear and electric fields: Deformation and burst of fluid drops, Proc. Roy. Soc. Lond. A, 267, 45–61, (1962). Arp, P. A., Foister, R. T., and Mason, S. G., Some electrohydrodynamic effects in fluid dispersions, Adv. Colloid Interface Sci., 12, 295–356, (1980). Garton, C. G., and Krasucki, Z., Bubbles in insulating liquids: Stability in an electric field, Proc. Roy. Soc. Lond. A, 280, 211–226, (1964).

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Griffiths, D. J., Introduction to Electrodynamics, Prentice-Hall, Upper Saddle River, NJ, (1989). Ha, J.-W., and Yang, S.-M., Deformation and breakup of Newtonian and non-Newtonian conducting drops in an electric field, J. Fluid Mech., 405, 131–156, (2000). Lu, Y., Electrohydrodynamic deformation of water drops in oil with an electric field, M.Sc. Thesis, University of Alberta, Edmonton, Canada, (2002). Melcher, J. R., and Taylor, G. I., Electrohydrodynamics: A review of the role of interfacial shear stresses, Ann. Rev. Fluid Mech., 1, 111–146, (1969). O’Konski, C. T., and Thatcher Jr., H. C., The distortion of aerosol droplets by an electric field, J. Phys. Chem., 57, 955–958, (1953). Taylor, G. I., Disintegration of water drop in an electric field, Proc. Roy. Soc. Lond. A, 280, 383–390, (1964). Taylor, G. I., Studies in electrohydrodynamics. The circulation produced by a drop in electric field, Proc. Roy. Soc. Lond. A, 291, 159–166, (1966). Torza, S., Cox, R. G., and Mason, S. G., Electrohydrodynamic deformation and burst of liquid drops, Phil. Trans. R. Soc. Lond. A, 269, 295–319, (1971). Vizika, O., and Saville, D. A., The electrohydrodynamic deformation of drops suspended in liquids in steady and oscillatory electric fields, J. Fluid Mech., 239, 1–21, (1992). Zholkovskij, E. K., Masliyah, J. H., and Czarnecki, J., An electrokinetic model of drop deformation in an electric field, J. Fluid Mech., 472, 1–27, (2002).

CHAPTER 5

ELECTRIC DOUBLE LAYER

In Chapter 3 we have dealt with the basics of electrostatics as applied to dielectrics. By and large, we did not deal with physical problems associated with free charges, i.e., ions, their spatial distribution with respect to a charged surface, and their mobility under an applied electric field. Knowledge of the spatial distribution of the free charges and their mobility forms the foundation towards the understanding of liquid flow, particle movement, and induced potential within the subject umbrella of electrokinetics. We define electrokinetics as the study of movement or flow under the influence of an electric potential and field. 5.1 ELECTRIC DOUBLE LAYERS AT CHARGED INTERFACES So far, we have reviewed the governing equations for the electric field distribution and the corresponding electrostatic potential caused by the presence of charges in vacuum and in dielectric materials. The problems of interest in dispersions require additional considerations. In particular, in aqueous systems of interest, one needs to develop methods to deal with the redistribution of ions in the solution caused by the presence of charged surfaces such as a charged particle, surface, or microemulsion droplet. The free ions in the solution are either attracted to or repelled from a charged surface depending on the sign of the surface charges. Such a redistribution of free ions in the solution together with the surface ions give rise to what are known as electric double layers. The purpose of the following sections of this chapter is to develop governing equations for the electric double layers and the double Electrokinetic and Colloid Transport Phenomena, by Jacob H. Masliyah and Subir Bhattacharjee Copyright © 2006 John Wiley & Sons, Inc.

105

106

ELECTRIC DOUBLE LAYER

layer potentials and to develop the forces of interaction between two such electric double layers. Before proceeding to discuss electric double layers, we shall briefly review the origin of the charge at interfaces and present an overview of the theoretical ideas. 5.1.1

Origin of Interfacial Charge

Most substances acquire a surface electric charge when brought into contact with an aqueous medium (Everett, 1988; Probstein, 2003). The stability of colloidal dispersions is very sensitive to the addition of electrolytes and to the surface charge of the colloidal particles. Direct evidence for the existence of charge on particles comes from the phenomenon of particle movement under an applied electric field (electrophoresis) which will be dealt with at a later stage. Although, in the course of electrokinetic studies, we accept the presence of surface charges and may pay less attention to their origin, it is still important to recognize the origin of these charges. Surfaces may become electrically charged by a variety of mechanisms. Some of the important mechanisms (Hunter, 1981; Everett, 1988; Lyklema, 1995) are listed below. 1. Ionization of Surface Groups. If a surface contains acidic groups, their dissociation gives rise to a negatively charged surface as shown in Figure 5.1(a). Conversely, a basic surface takes on a positive charge, Figure 5.1(b). In both cases, the magnitude of the surface charge depends on the acidic or basic strengths of the surface groups and on the pH of the solution. The surface charge can be reduced to zero at the point of zero charge, PZC, by suppressing the surface ionization. This can be achieved by decreasing the pH for the case of a basic surface. Most metal oxides can have either a positive or a negative surface charge depending on the bulk pH. 2. Differential Dissolution of Ions from Surfaces of Sparingly Soluble Crystals. For example, when a silver iodide crystal (AgI) is placed in water, dissolution occurs until the product of ionic concentration equals the solubility product [Ag+ ][I− ] = 10−16 (mol/L)2 . If equal amounts of Ag+ and I− ions were to dissolve, then [Ag+ ] = [I− ] = 10−8 (mol/L)2 and the surface would

COOH

COO

COOH

COO

COOH

COO (a)

H+ H+ H+

OH

+

OH

+

OH

+

OH OH OH

(b)

Figure 5.1. Acquisition of surface charge by ionization of (a) acidic groups and (b) basic groups at a surface.

5.1 ELECTRIC DOUBLE LAYERS AT CHARGED INTERFACES

107

be uncharged. However, silver ions dissolve preferentially, leaving a negatively charged surface. If Ag+ ions now are added in the form of, say, a silver nitrate (AgNO3 ) solution, the preferential solution of silver ions is suppressed and the charge falls to zero at a particular concentration. Further addition of AgNO3 leads to a positively charged surface since it is now iodide ions that are preferentially dissolved. 3. Isomorphic Substitution. Clays may exchange an adsorbed intercalated or structural ion with one of lower valency, thus producing a negatively charged surface. For example,Al3+ may replace Si4+ in the surface of the clay, producing a negative surface charge. In this case, a point of zero surface charge can be reached by reducing the pH. Here, the added H+ ions combine with the negative charges on the surface to form OH groups. 4. Charged Crystal Surfaces. It may happen that when a crystal is broken, surfaces with different properties are exposed. Thus, in some clays (e.g., kaolinite), when a platelet is broken, the exposed edges contain Al(OH)3 groups which take up H+ ions to give a positively charged edge. This edge surface charge (positive) may coexist with negatively charged basal surfaces, leading to special properties. In this case, there will be no single PZC, but each type of surface will have its own characteristic PZC. For kaolinite, the flat surface is negatively charged and the edges are positively charged at low pH. At high pH, the positive charge on the edges decreases. 5. Specific Ion Adsorption. Surfactant ions may be specifically adsorbed on surfaces. Cationic surfactants can adsorb to negatively charged surfaces to yield net positive charges on the surface, while anionic surfactants can mask the positive charge of a surface by adsorbing onto them. The modification of surface charge by the adsorption of surfactants is shown in Figure 5.2. Surfactants play a major role in modifying surface charges and consequently affect the behavior of colloidal particles in terms of their stability. For instance, surfactants already present in crude oils play an important role in oil recovery. In all cases, the pH of the electrolyte solution into which a surface is immersed affects the surface electric potential.

R-NH + 3

R-NH3Cl R-NH3Cl

R-NH + 3

R-NH3Cl

R-NH + 3

R-NH3Cl (a)

Cl–

– R-SO 3

R-SO3H

– Cl– Cl

R-SO3H

R-NH + 3 Cl–

R-SO3H

– R-SO 3 – R-SO 3

R-SO3H

H+ H+ – R-SO 3 + H H+

(b)

Figure 5.2. Acquisition of surface charge by adsorption of (a) cationic surfactants and (b) anionic surfactants at a surface.

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ELECTRIC DOUBLE LAYER

5.1.2

Electrical Potential Distribution Near an Interface

When electrolytes are present in water under no-flow conditions, for a sufficiently large representative volume, (i.e., a volume that is several orders of magnitude larger than the volume of the ions and the water molecules), and at locations far away from the container’s walls, electroneutrality condition is obeyed in an average sense. In other words, the sum of all the charges within the representative volume due to the ions is zero. The presence of electroneutrality is quite intuitive as there is no preferential spatial distribution of the ions in the bulk solution. The Brownian motion is sufficient to create a homogeneity in the spatial distribution of the ions. However, when a surface, say a glass slide is immersed in an electrolyte solution, the glass surface acquires a surface charge. Although the origin of this surface charge has been discussed earlier, in the following analysis, the reason as to why the glass surface attains a surface charge is not of immediate interest to us. For the present, we will assume that the glass surface is charged. Clearly, the charged surface will influence the distribution of the nearby ions in the electrolyte solution. Ions of opposite charge to that of the surface, called counterions, will be attracted toward the surface, while ions of like charge, called coions, will be repelled from the surface.1 As a consequence of the attraction and repulsion between the ions in the electrolyte solution and the charged surface, there will be a non-uniform ionic distribution normal to the surface. Intuitively, one can state that there will be a higher concentration of counterions near the surface and that at a distance sufficiently far away from the charged surface, ionic electroneutrality will be reestablished. However, the manner by which the counterions and coions distribute themselves near and far away from the charged surface and the extent of the influence of the charged surface on their spatial distribution need some elaboration. From the above discussion, the presence of a charged surface in an electrolyte solution will influence the ion distribution close to it. Such a redistribution of the free ions in solution gives rise to what we call the electric double layer. In simple terms, its name came about because of the separation of charge between the surface and the electrolyte solution. One layer is the charge on the surface, and the other, a “layer” of ions in the vicinity of the surface. The concept of the electric double layer was introduced by Helmholtz, who envisaged an arrangement of charges in two parallel planes as shown in Figure 5.3(a), forming, in effect, a “molecular condenser”. However, thermal motion causes the counterions to be spread out in space, forming a diffuse double layer, as shown in Figure 5.3(b). The theory for such a diffuse double layer was developed independently by Gouy and Chapman in the early 1900s (Hunter, 1981). In the Gouy–Chapman diffuse double layer model, the charged surface, composed of one layer of charges, has a surface potential ψs . The compensating ions in solution are regarded as point charges immersed in a continuous dielectric medium. The repulsion/attraction coupled with the random thermal Brownian motion of the ions within the dielectric medium gives rise to a diffuse electrical layer. Within this diffuse layer, there is no charge neutrality. 1

The term counterions is used for the ions which have the opposite sign to the surface charge. The term coions is used for the ions which have the same sign as the surface charge.

5.1 ELECTRIC DOUBLE LAYERS AT CHARGED INTERFACES

109

Figure 5.3. The electric double layer. (a) according to the Helmholtz model, (b) the diffuse double layer resulting from thermal motion.

The equilibrium concentration distribution of ions in the diffuse layer is established due to the forces attributed to electrostatic attraction/repulsion between the charged surface and the ions, and diffusion of ions due to concentration gradients. The Gouy–Chapman model provides good quantitative predictions when the surface potential is low (∼0.025 V) and the electrolyte concentration is not too high. A major defect of the model is that it neglects the finite size of the ions (it assumes that the ions are point charges that can approach the surface without limit) and the non-ideality of the solution. Moreover, the dielectric permittivity of the medium is taken to be constant all the way to the surface. A modification to the Gouy–Chapman model is offered by Stern (1924) and it will be elaborated upon at a later stage. 5.1.3 The Boltzmann Distribution We have mentioned in the previous section, that due to the presence of a charged surface, there will be a spatial distribution of the ions normal to the surface. Of significance, there is equilibrium of the ionic distribution within the diffuse electric double layer. Such an equilibrium would allow one to use Boltzmann distribution to relate the ionic concentration to the electric potential. This type of a relationship is needed in order to analyze the ionic spatial distributions in electric double layers. The Boltzmann distribution is a fundamental relation in statistical thermodynamics that describes the probability of the occurrence of microscopic states as a function of the energy of these states. Here, we shall introduce Boltzmann’s distribution using simple concepts. However, at a later stage, a more formal derivation will be given. Whenever thermodynamic equilibrium exists, the probability that the system energy is confined within the range W and W + dW is proportional to dW , and can be represented as P(W )dW . The function P(W ), which is referred to as the probability density, is given by
 W P ∝ exp − (5.1) kB T

110

ELECTRIC DOUBLE LAYER

as shown by Boltzmann. In Eq. (5.1), T is the absolute temperature (K), kB is the Boltzmann constant (J/K) given by R/NA , where R is the universal gas constant and NA is Avogadro’s number. The above expression follows from statistical considerations (see, for example, Kittel and Kroemer 1980; McQuarrie, 1976). In the present context, W represents the “energy” corresponding to a particular location of an ion (defined relative to a suitable reference state). The appropriate choice here is the work W required to bring one ion of valency zi (i.e., a charge zi e) from infinity, where ψ = 0, to a given location x, having a potential ψ, given by W = zi eψ. Therefore, the probability density of finding an ion at location x can be written as2
 zi eψ (5.2) P ∝ exp − kB T Similarly, the probability density of finding the ion at the neutral state (ψ = 0) is P0 ∝ exp(0)

(5.3)

For convenience, we take the neutral state to be at ψ = 0. The ratio of P to P0 is taken as being equal to the ratio of the concentrations of the species i at the respective states. Combining Eqs. (5.2) and (5.3) leads to
 zi eψ ni = ni∞ exp − kB T

(5.4)

where ni∞ is the ionic number concentration at the neutral state where ψ = 0 and ni is the ionic number concentration of the i th ionic species at the state where the electric potential is ψ. The ionic valency zi can be either positive or negative depending on whether the ion is a cation or an anion, respectively. As an example, for the case of CaCl2 salt, z for the calcium ion is +2 and it is −1 for the chloride ion. The Boltzmann distribution defined in Eq. (5.4) is of fundamental importance. It relates the ionic number concentration, ni , of the i th species at a given location to the electric potential at that location. The Boltzmann distribution will be employed in evaluating the spatial variation of the electric potential and the ionic number concentration in the diffuse electric double layer. EXAMPLE 5.1 The Barometric Equation. Use the concept of the Boltzmann distribution to develop an equation for the density of a gas in the atmosphere as a function of altitude. 2

A more exact probability density function is given as
 Wi P = exp − kB T

where Wi = ezi ψ + Wri , with Wri being the energy associated with reorganization of the medium due to the introduction of the ion. We assume here that Wri /kB T ≪ 1.

5.2 POTENTIAL FOR PLANAR ELECTRIC DOUBLE LAYER

111

Solution Consider the concentration of, say, helium in the atmosphere as a function of distance (height) from the earth’s surface. The potential energy is given by mgx, where m is the mass of a helium atom, g is the acceleration due to gravity, and x is the distance from the earth’s surface. The probability density, P(x), of finding a helium atom at an altitude between x and x + dx is given by
 −mgx P(x) = α exp kB T The constant of proportionality α is independent of x. At zero altitude, the probability density is 
0 =α P(0) = α exp kB T The above probability densities are proportional to the concentration n(x) of He (i.e., the number of He atoms per unit volume) at the appropriate altitudes. Therefore, the ratio of these probability densities is equal to the ratio of the concentrations: n(x) P(x) = , P(0) n(0)

(5.5)


 mgx n(x) = n(0) exp − . kB T

(5.6)

i.e.,

Clearly, the density n(x) can also be expressed in other units, e.g., in mol/m3 . Equation (5.6) is related to the barometric equation, which describes the pressure variation with altitude at constant temperature (see Hiemenz and Rajagopalan, 1997).

5.2 POTENTIAL FOR PLANAR ELECTRIC DOUBLE LAYER In Chapter 3, we showed that the electric potential in a dielectric carrying free charge is governed by the Poisson equation, Eq. (3.116). In an electrolyte solution, the continuous phase is water which is a dielectric medium. The free charges are the ions contained in the electrolyte solution. Consequently, Poisson’s equation is also the appropriate equation to be employed to analyze the electric diffuse double layers where the dielectric permittivity of water is assumed constant. 5.2.1

Gouy–Chapman Analysis

In order to facilitate the analysis, the case of a flat surface will be considered. The objective here is to obtain analytical expressions for the distribution of potential and

112

ELECTRIC DOUBLE LAYER

Charged surface

=

Electrolyte solution

s

0

x Distance away from the surface,

Figure 5.4. Potential distribution near a flat surface.

ion concentrations due to the presence of a charged surface in a dielectric medium having free charge. The Poisson equation is given by ǫ∇ 2 ψ = −ρf

(5.7)

For a one-dimensional problem as shown in Figure 5.4, the above equation simplifies to d 2ψ (5.8) ǫ 2 = −ρf dx where x is the distance normal to the charged surface. The space charge density of the mobile (“free”) ions, ρf , can be written in terms of the number concentrations of the ions and the corresponding valencies as

ρf =

N 

zi eni

(5.9)

i=1

where ni is the ionic number concentration of the i th species (say, in m−3 ), zi is the valence of the i th ionic species (with the appropriate sign), e is the magnitude of the fundamental (elementary) charge on an electron, 1.602 × 10−19 C, and N is the number of ionic species in the electrolyte solution. Equations (5.8) and (5.9) may now be combined to give ǫ

N  d 2ψ = − zi eni dx 2 i=1

(5.10)

We can now write the right-hand side of the above equation in terms of ψ by relating the spatial distribution of the ions to ψ using the Boltzmann distribution. Substituting for ni in the above equation through the use of the Boltzmann distribution given by

5.2 POTENTIAL FOR PLANAR ELECTRIC DOUBLE LAYER

113

Eq. (5.4) leads to the well-known Poisson–Boltzmann equation ǫ


 N  d 2ψ zi eψ = − z en exp − i i∞ dx 2 kB T i=1

(5.11)

The Poisson–Boltzmann equation defines the electric potential distribution in the diffuse ionic layer adjacent to a charged surface subject to appropriate boundary conditions. In order to facilitate the solution of Eq. (5.11), the special case of a single salt dissociating into cationic and anionic species (i.e., N = 2) will be considered with the added simplification of symmetric (z : z) electrolyte solution (e.g., NaCl, CuSO4 , or AgI). In symmetric electrolytes, both the cations and anions have the same valencies. In the case of planar electric double layers, one can obtain an analytical solution for ψ for symmetric electrolytes without any further approximations, and such a solution is known as the Gouy–Chapman theory. In particular, the Gouy–Chapman analysis does not require the linearization of the Boltzmann approximation and is, therefore, a nonlinear theory. We shall consider the linearization in the following subsection. For a symmetric electrolyte, one can write z+ = −z− = z where z is the valence of the cation. Equation (5.11) can then be written as 

 zeψ zeψ d 2ψ − exp ǫ 2 = −zen∞ exp − dx kB T kB T

(5.12)

(5.13)

or


 d 2ψ zeψ ǫ 2 = 2zen∞ sinh dx kB T

(5.14)

where n+∞ = n−∞ = n∞ , the ionic number concentration in the bulk solution where ψ = 0. The appropriate boundary conditions for Eq. (5.14) are x=0 x→∞

ψ = ψs

ψ =0

(5.15-a) (5.15-b)

where ψs is the surface potential at x = 0. The solution to Eq. (5.14) under the above conditions is   1 + exp(−κx) tanh(s /4) (5.16)  = 2 ln 1 − exp(−κx) tanh(s /4)

where  is the dimensionless potential defined as =

zeψ kB T

(5.17)

s =

zeψs kB T

(5.18)

from which, one has

114

ELECTRIC DOUBLE LAYER

In Eq. (5.16), κ −1 is the Debye length, which is defined as κ −1 =




ǫkB T 2e2 z2 n∞

1/2

(5.19)

The Debye length, κ −1 , is a measure of the electric double layer thickness, and is a property of the electrolyte solution. It should be noted that this parameter contains information about the dielectric permittivity of the solvent, ǫ, as well as the valence, z, and bulk concentration, n∞ , of the ions. However, no information regarding the properties of the charged surface is present in the Debye length. Although it is normally referred to as the thickness of the electric double layer, the actual thickness of a double layer extends well beyond κ −1 . Typically, the Debye length represents a characteristic distance from the charged surface to a point where the electric potential decays to approximately 33% of the surface potential. For s ≪ 1, i.e., low surface potentials, Eq. (5.16) can be approximated to   1 + 0.25s exp(−κx)  = 2 ln (5.20) 1 − 0.25s exp(−κx) Figure 5.5 shows a comparison between the exact solution given by Eq. (5.16) and the approximate solution given by Eq. (5.20). For s = 0.5, both the expressions yield nearly identical distributions of  but, as would be expected, the two results differ for the higher surface potential of s = 2. As it will be shown later, Eq. (5.20) gives a poorer approximation for the potential distribution than the Debye–Hückel approximation, Eq. (5.24), for s > 1.

Figure 5.5. Potential distribution near a flat surface. Comparison of exact and approximate solutions.

5.2 POTENTIAL FOR PLANAR ELECTRIC DOUBLE LAYER

115

EXAMPLE 5.2 The Debye Thickness for Symmetric Electrolytes. Calculate the Debye length for a number of values of the ionic strength for a symmetric electrolyte. Solution The condition of electroneutrality is usually assumed to occur far away from the surface and is given as  z i ni = 0 For a binary electrolyte, one has

z− n− + z+ n+ = 0 which, for a symmetric (z : z) electrolyte, leads to n− = n+ = n∞ The Debye length κ −1 for a symmetric electrolyte then follows from κ

−1

=




ǫkB T 2e2 z2 n∞

 21

where e = 1.602 × 10−19 C and for water at T = 298 K, ǫ = 6.95 × 10−10 C2 N−1 m−2 (the value of ǫr = 78.5). Now, n∞ , the ionic number concentration, is given by 


1 mol L × 1000 3 NA n∞ = M L m mol or n∞ = 1000NA M, with the Avogadro number NA = 6.022 × 1023 mol−1 and M being the molar concentration (mol/L) of the electrolyte. The expression for the Debye length then becomes 3.04 κ −1 = √ × 10−10 m z M The ionic strength, I , of an electrolyte solution is defined as I=

1 2 z Mi 2 i i

where Mi is the molar concentration of the i th ionic species arising from the dissociation of the electrolyte and zi is its valence. For a 0.1 M (1 : 1) electrolyte (such as NaCl), M+ = M− = 0.1 M. Hence, the ionic strength is given by I=

 1 2  1 2 2 z+ M+ + z− M− = (1) × 0.1 + (1)2 × 0.1 = 0.1 M 2 2

116

ELECTRIC DOUBLE LAYER

Thus, for a symmetric (1 : 1) electrolyte, the ionic strength is equal to its molarity. Although the term ionic strength is frequently used in electrochemistry to denote electrolyte concentration, we will use molar and number concentrations in this book. The values of the Debye length, κ −1 , for different electrolyte concentrations for the case of z = 1 are shown in the following table. In this case, the ionic strength is equal to the molar concentration of the electrolyte. It is clear from the tabulated results that κ −1 decreases as the electrolyte concentration increases. At high molarity, the electric double-layer thickness becomes very small. In a non-electrolyte system, however, the double-layer thickness can be thought of as extending to infinity (i.e., a large distance from the surface). Ionic Concentration, M 10−6 10−4 10−2

5.2.2

Debye Length κ −1 , nm 304.0 30.4 3.04

Debye–Hückel Approximation

When the surface potential is small, say, ψs ≪ 0.025 V (Hiemenz and Rajagopalan, 1997), the term zeψ/kB T is smaller than unity and one can approximate the hyperbolic sine function as follows:
 zeψ zeψ zeψ sinh for ≪1 (5.21) ≈ kB T kB T kB T Making use of the above approximation and the definition of κ given by Eq. (5.19), the Poisson–Boltzmann equation (5.14) becomes d 2ψ 2e2 z2 n∞ = ψ = κ 2ψ 2 dx ǫkB T

(5.22)

Equation (5.22) is the linearized version of the Poisson–Boltzmann equation and is one of the most commonly used equations in the electric double layer theory (even when linearization may not be justifiable!). The linearization of the Poisson– Boltzmann equation in the above manner is generally referred to as the Debye–Hückel approximation. Its solution, with the boundary conditions ψ = ψs at x = 0 and ψ = 0 at x → ∞, follows readily and is given by ψ = ψs exp (−κx)

(5.23)

 = s exp (−κx)

(5.24)

or, in dimensionless form by

The dimensionless surface potential, s , is defined in Eq. (5.18). Equation (5.24) indicates that the potential decays exponentially as one moves away from the charged surface. It may be noted here that Eq. (5.20) can be simplified to yield

5.2 POTENTIAL FOR PLANAR ELECTRIC DOUBLE LAYER

117

Figure 5.6. Potential distribution near a flat surface. Comparison between exact and Debye– Hückel solutions.

the Debye–Hückel potential distribution, Eq. (5.24). A comparison between the exact solution provided by Eq. (5.16) and the Debye–Hückel approximate solution, Eq. (5.24) is given in Figure 5.6 for two values of surface potentials, namely, s = 0.5 and s = 2. For the case of the low surface potential, s = 0.5, both the exact and Debye–Hückel solutions give essentially the same dimensionless electric potential variation with κx. For the higher surface potential of s = 2, there is only a slight difference between the two profiles. The Debye–Hückel solution normally gives fairly close /s variations for s values as high as 3. Comparing Figures 5.5 and 5.6 it becomes evident that Eq. (5.20) is less accurate than the Debye–Hückel expression, Eq. (5.24) as κx → 0 for large surface potentials, s = 2. Tabulation of the Debye length (κ −1 ) for different electrolyte concentrations and valence types is given in Table 5.1. The Debye double layer thickness, κ −1 , varies from 9.61 × 10−9 m for a molarity of 0.001 M for a 1 : 1 electrolyte to 0.32 × 10−9 m for a 3 : 3 electrolyte at a molarity of 0.1. Increasing the molarity of the electrolyte and its valency tends to decrease the value of the Debye length κ −1 . The variation of /s with distance is shown in Figure 5.7 for a 1 : 1 electrolyte at three concentrations using the Debye–Hückel solution. Increasing the electrolyte concentration leads to a fast decay in /s and a small electric double layer thickness. The values of κ −1 are indicated by solid circles. The variation of /s with distance at 0.001 M concentration for symmetric electrolytes of three different valencies is shown in Figure 5.8 where the Debye–Hückel approximation is used. The decay in /s is much sharper for the 3 : 3 electrolyte solution when compared to the lower valency electrolytes. As well, the electric double layer thickness (Debye length) is smaller for the 3 : 3 electrolyte. One can conclude from Figure 5.8 that for a given electrolyte concentration, the higher valency electrolytes are more effective in altering the electric potential inside the electric double layer. Consequently, higher electrolyte concentrations and electrolytes having higher valencies tend to “screen” the electric potential due to a charged surface to a larger extent.

118

ELECTRIC DOUBLE LAYER

TABLE 5.1. Values of κ and κ −1 for Several Different Electrolyte Concentrations and Valences with Numerical Formulas for these Quantities Given for Aqueous Solutions at 25◦ C. Symmetrical Electrolyte Molarity 0.001

0.01

0.1

z+ : z−

κ (m ) = 3.29 × 109 × |z|M 1/2

κ −1 (m) = 3.04 × 10−10 × |z|−1 M −1/2

1:1 2:2 3:3 1:1 2:2 3:3 1:1 2:2 3:3

1.04 × 108 2.08 × 108 3.12 × 108 3.29 × 108 6.58 × 108 9.87 × 108 1.04 × 109 2.08 × 109 3.12 × 109

9.61 × 10−9 4.81 × 10−9 3.20 × 10−9 3.04 × 10−9 1.52 × 10−9 1.01 × 10−9 9.61 × 10−10 4.81 × 10−10 3.20 × 10−10

−1

Asymmetrical Electrolyte Molarity 0.001

0.01

0.1

z+ : z− 1 : 2, 2 : 1 1 : 3, 3 : 1 2 : 3, 3 : 2 1 : 2, 2 : 1 1 : 3, 3 : 1 2 : 3, 3 : 2 1 : 2, 2 : 1 1 : 3, 3 : 1 2 : 3, 3 : 2

κ (m ) = 2.32 × 109  2 1/2 × i zi Mi −1

1.80 × 108 2.54 × 108 4.02 × 108 5.68 × 108 8.04 × 108 1.27 × 108 1.80 × 109 2.54 × 109 4.02 × 109

κ −1 (m) = 4.30 × 10−10  2 −1/2 × i zi Mi 5.56 × 10−9 3.93 × 10−9 2.49 × 10−9 1.76 × 10−9 1.24 × 10−9 7.87 × 10−10 5.56 × 10−10 3.93 × 10−10 2.49 × 10−10

EXAMPLE 5.3 Poisson–Boltzmann Equation for Flat Surface. Derive the Poisson–Boltzmann equation for a flat charged surface immersed in an arbitrary electrolyte solution. Use the Debye–Hückel approximation. Solution The Poisson equation is given by Eq. (5.8) and it is written as ǫ

d 2ψ = −ρf dx 2

(5.25)

where x is the Cartesian coordinate normal to the surface. Making use of the definition of ionic free charge density, ρf , one can write ρf =

N  i=1

ni z i e

(5.26)

5.2 POTENTIAL FOR PLANAR ELECTRIC DOUBLE LAYER

119

Figure 5.7. Normalized double layer potential versus distance from a surface according to the Debye–Hückel approximation. Curves drawn for a 1 : 1 electrolyte at three concentrations.The solid circles represent the values of κ −1 corresponding to each concentration.

where N is the total number of ionic species in the system. Substituting Eq. (5.26) in Eq. (5.25), one obtains ǫ

N  d 2ψ = − ni z i e dx 2 i=1

(5.27)

Figure 5.8. Normalized double layer potential vs. distance from a surface according to the Debye–Hückel approximation. Curves drawn for 0.001 M symmetrical electrolytes of three different valence types. The solid circles represent the values of κ −1 corresponding to each valence type.

120

ELECTRIC DOUBLE LAYER

The ionic concentration is provided by the Boltzmann equation as 
−zi eψ ni = ni∞ exp kB T

(5.28)

Combining Eqs. (5.27) and (5.28) leads to ǫ


 N  d 2ψ −zi eψ = −e n z exp i∞ i dx 2 kB T i=1

Invoking the Debye–Hückel approximation, where    zi eψ    k T ≪1 B

(5.29)

(5.30)

and recognizing that Eq. (5.29) becomes

exp(−y) ≈ 1 − y

for

y≪1


 N   zi eψ d 2ψ ǫ 2 = −e ni∞ zi 1 − dx kB T i=1

(5.31)

(5.32)

Upon rearranging, Eq. (5.32) becomes  N   zi2 ni∞ eψ d 2ψ ni∞ zi − ǫ 2 = −e dx kB T i=1

(5.33)

From the condition of electroneutrality, we have N  i=1

ni∞ zi = 0

(5.34)

which simplifies Eq. (5.33) to N

ǫ

 z2 e2 ni∞ d 2ψ i =ψ 2 dx kB T i=1

(5.35)

Letting κ2 = or κ2 =

N  z2 e2 ni∞ i

i=1

ǫkB T

N e2  2 z ni∞ ǫkB T i=1 i

(5.36)

121

5.2 POTENTIAL FOR PLANAR ELECTRIC DOUBLE LAYER

The Poisson–Boltzmann equation for a planar surface at low potentials becomes d 2ψ = κ 2ψ dx 2

(5.37)

Equation (5.37), which describes the electrical potential in the electric double layer, is valid for an arbitrary electrolyte and it is not limited to the special case of a symmetric electrolyte as long as the more general definition of κ, given by Eq. (5.36), is employed. For a symmetric electrolyte (z : z), Eq. (5.36) reverts to Eq. (5.19). EXAMPLE 5.4 Debye Length. In Table 5.1, the Debye length was given for an asymmetrical electrolyte in water at 298 K as κ −1 = 4.30 × 10−10



zi2 Mi

i

−1/2

where zi is the valence of the i th ionic species and Mi is its molarity. Derive the above expression. Solution For the general case of asymmetric electrolytes, the Debye length (thickness) is given by Eq. (5.36) of Example 5.3, where

κ

−1

=



Now ni∞ = Mi

N e2  2 z ni∞ ǫkB T i=1 i

−1/2

(5.38)

mol 1000 L 1 · NA · L m3 mol

or ni∞ = 1000Mi NA

(5.39)

Substituting for ni∞ from Eq. (5.39) into Eq. (5.38) leads to κ −1 =



N 1000NA e2  2 zi Mi ǫkB T i=1

−1/2

(5.40)

Inserting known parameter values, Eq. (5.40) gives

κ

−1

=



N 1000 × 6.022 × 1023 × (1.602 × 10−19 )2  2 z Mi 6.95 × 10−10 × 1.381 × 10−23 × 298 i=1 i

−1/2

(5.41)

122

ELECTRIC DOUBLE LAYER

Equation (5.41) leads to

κ

−1

= 4.30 × 10

−10

N  i=1

zi2 Mi

−1/2

(5.42)

The expression of the Debye length as given above is useful when one deals with the general case of a mixture of electrolytes in a solution. EXAMPLE 5.5 Debye Length for a Mixture. A 400 ml of 0.01 M sodium chloride solution is mixed with 600 ml of 0.001 M sodium sulfate. What is the Debye length for this electrolyte solution? Solution The molarity of NaCl upon mixing becomes 0.01 ×

400 = 0.004 M (400 + 600)

The molarity of Na2 SO4 upon mixing becomes 0.001 ×

600 = 0.0006 M (400 + 600)

Now, we can write for NaCl and Na2 SO4  



zi2 Mi = 12 × 0.004 + 12 × 0.004 + 12 × 0.0006 × 2 + 22 × 0.0006 = 0.008 + 0.0036 = 0.0116

Now, substituting this value in Eq. (5.42) yields κ −1 = 4.30 × 10−10 × (0.0116)−1/2 = 3.99 × 10−9 m 5.2.3

Surface Charge Density

We have made use of both the Boltzmann distribution and the Poisson equation to investigate the electric potential distribution inside the electric double layer for a planar surface. We now seek to relate the electric surface potential to the surface charge density. Let us consider a planar surface being placed in an electrolyte solution. As the surface is assumed to be electrically neutral prior to its placement in the solution, the acquired charge must balance the ionic charge within the electric double layer. This ionic balance is required due to the overall electroneutrality of the electrolyte solution and the surface. In other words, ionic charge conservation must be respected.

5.2 POTENTIAL FOR PLANAR ELECTRIC DOUBLE LAYER

It follows then that for a planar surface  qs = −

∞ 0

ρf dx

123

(5.43)

where x is the coordinate normal to the surface, qs is the surface charge density (C/m2 ), and ρf is the free charge density, i.e., ionic charge density, C/m3 . Recalling the Poisson equation for one dimension, ǫ

d 2ψ = −ρf dx 2

one can integrate Eq. (5.44) to give
  ∞  ∞ d dψ ǫ ρf dx dx = − dx dx 0 0 Combining Eqs. (5.43) and (5.45) and evaluating the integral yields   dψ ∞ qs = ǫ dx 0

(5.44)

(5.45)

(5.46)

Expanding Eq. (5.46) gives qs = ǫ



   dψ  dψ  − dx ∞ dx 0

(5.47)

At large distances away from the surface, x → ∞, the electric potential is taken as zero and Eq. (5.47) becomes  dψ  qs = −ǫ (5.48) dx  0

Equation (5.48) states that the surface charge density is proportional to the gradient of the electric potential at the charged surface. The proportionality constant is simply the medium permittivity. A more general form of Eq. (5.48) is given as qs = −ǫ∇ψ · n

(5.49)

where n is the outward normal to a charged surface. Recall that the electric field strength is equal in absolute magnitude to the electric potential gradient. Therefore, for a one-dimensional case, one can write at x = 0  dψ  (5.50) Ex |0 = − dx 0 Considering Eqs. (5.48) and (5.50) and letting Ex |0 ≡ Es , one can write qs = ǫEs qs Es = ǫ

(5.51-a) (5.51-b)

124

ELECTRIC DOUBLE LAYER

Equations (5.51-a) or (5.51-b) relate the surface charge density to the electric field strength at the surface. Equation (5.51-a) is a statement of the Gauss law. The evaluation of the surface charge density, qs , as provided by Eq. (5.48) would depend on the choice of the electric potential expression to be used. Let us make use of Gouy–Chapman solution, Eq. (5.16), which is valid for a symmetric electrolyte and is not limited to small surface potentials. Differentiating ψ of Eq. (5.16) with respect to x and substituting for dψ/dx in Eq. (5.48) gives qs = 2[2ǫkB T n∞ ]1/2 sinh(s /2)

(5.52)

The dimensionless surface potential is defined in Eq. (5.18). For the special case of small surface potential s ≪ 1, one can simplify Eq. (5.52) to qs = [2ǫkB T n∞ ]1/2 s

(5.53)

In terms of the dimensional surface potential, Eq. (5.52) gives for s ≪ 1 qs = ǫκψs

(5.54)

where κ is given by Eq. (5.19). Equation (5.52) relates the surface charge density to its potential for a planar surface. Rather interesting information can be extracted from this relationship. In real systems, it is the surface charge that ultimately controls the surface potential. Consequently, in terms of the electrolyte molarity, one can rewrite Eq. (5.52) as sinh(s /2) =

(qs /e) (2/e)[2000ǫkB T NA M]1/2

(5.55)

The term (qs /e) can be thought of as the ionic surface charge density, i.e., ions/m2 . Equation (5.55) indicates that the surface electric potential increases with the ionic surface charge density (qs /e) and decreases with increasing electrolyte molarity. Figure 5.9 shows the variation of the surface electric potential, s , with (qs /e)∗ for different electrolyte molarities, where qs∗ is the surface charge in C/nm2 .A vertical line at (qs /e)∗ = 0.2, which represents one charge per 5 nm2 (a reasonably ionized surface) is drawn on Figure 5.9. If the charged surface maintains its surface ionization at all electrolyte concentrations, the intersection of the the vertical dashed line with the individual solid lines would then represent the surface potential at the different electrolyte concentrations. In real systems, however, surface ionization is a function of electrolyte concentration and pH. Therefore, to properly predict the surface potential one would need additional information on surface ionization and surface charge density characteristics. We will discuss models employed to predict surface potentials later in this chapter. In summary, for the general case, the Gouy–Chapman model relates the surface charge density to the surface potential through Eq. (5.52). For small surface potentials, say

5.2 POTENTIAL FOR PLANAR ELECTRIC DOUBLE LAYER

125

Figure 5.9. Effect of surface charge density on the surface potential of a planar surface.

ψs < 0.025 V, Eq. (5.54) provides the link between the surface charge density and the surface electric potential. It is of interest to note that Eqs. (5.53) and (5.54) suggest the manner by which one can non-dimensionalize the surface charge density. Equation (5.53) suggests the use of (2ǫkB T n∞ )1/2 and Eq. (5.54) suggests ǫκ(kB T /ze). Utilizing the equivalence of these two terms, we may write n∞ kB T ǫ = 2 κ 2




kB T ze

2

(5.56)

It is noteworthy that each term in the above expression has units of force (Newtons in SI units). These combinations of parameters can be used interchangeably to scale the interaction energy or force arising from the overlap of two electric double layers. We will revisit these terms in the later sections of this chapter. 5.2.4

Ionic Concentrations in Electric Double Layers

As we discussed earlier, the presence of a charged surface disturbs the electroneutrality of the electrolyte solution in the immediate vicinity of the charged surface. Counterions are attracted to the surface and coions are repelled from the surface. The spatial distribution of the ions normal to the surface can be provided via the use of the Boltzmann distribution and an expression for the electric potential. Let us derive expressions for the variation of the ion concentration ratio along the coordinate normal to the surface. The Boltzmann distribution, Eq. (5.4) gives


zi eψ ni = ni∞ exp − kB T



(5.57)

126

ELECTRIC DOUBLE LAYER

Making use of the Debye–Hückel approximation, Eq. (5.24), the Boltzmann distribution, Eq. (5.57), for a symmetric (z : z) electrolyte provides n+ 2 = exp[−s exp(−κx)] ≃ 1 − s exp(−κx) + s exp(−2κx) + O(s3 ) n∞ 2 (5.58-a) 2 n− = exp[s exp(−κx)] ≃ 1 + s exp(−κx) + s exp(−2κx) + O(s3 ) n∞ 2 (5.58-b) where  = zeψ/kB T . Plots of Eqs. (5.58-a) and (5.58-b) are shown in Figure 5.10. The decay in the number concentration of the counterions is more steep when compared to that of the coions. In both cases, the maximum deviation from the bulk conditions occurs at the charged surface. The difference between the counterion and coion number concentrations is referred to as the excess ionic number concentration. A plot of the variation of normalized excess number concentration, (n− − n+ )/n∞ , with the distance away from the charged surface is shown in Figure 5.11. As would be expected, the excess ion concentration has its maximum value at the surface and it decays to zero far away from the surface where the electroneutrality condition holds. One can think of the term (n− − n+ )/n∞ as being a measure of the deviation from electroneutrality. Noting that for a symmetric electrolyte the net excess charge will be proportional to the difference in the ionic number concentration, (n− − n+ ), the normalized excess ionic concentration, (n− − n+ )/n∞ , can also be called the normalized excess charge. The variation of the normalized excess charge is shown in Figure 5.12 for different

⁄

⁄

Figure 5.10. Coion and counterion distribution near a charged planar surface.

127

(n - n+)/n

5.2 POTENTIAL FOR PLANAR ELECTRIC DOUBLE LAYER

Figure 5.11. Distribution of excess ion concentration normalized with respect to the bulk ion concentration near a planar surface.

values of surface electrical potential. It is clear that as the surface potential, s , increases, the normalized excess charge increases as well. Since both n− and n+ are positive quantities, such a variation in (n− − n+ )/n∞ would imply that n− becomes much larger than n+ with increasing surface potential, s . In this example, the surface potential was taken as positive and, hence, the ionic number concentration n− is that of the counterions.

Figure 5.12. Normalized excess ion concentration near a planar surface: Effect of surface potential.

128

ELECTRIC DOUBLE LAYER

The plot of Figure 5.12 indicates that the counterions become more dominant than the coions near the surface as the surface potential becomes high. Consequently, for high surface potentials, one may only need to consider the counterion concentrations near the charged surfaces in solving the Poisson equation. A further generalization can be made in that counterions are the dominant ions in determining the surface charge behavior. Consideration of counterions alone will be considered in the next section. 5.2.5

High Surface Potentials and Counterion Analysis

We have discussed two solutions for the electric potential distribution inside the diffuse double layer. First, we presented the Gouy–Chapman analysis where no assumption for low surface potential was made but its validity was limited to symmetrical electrolytes, i.e., (z : z) electrolytes. Secondly, we used Debye–Hückel approximation to obtain a solution that was valid for low surface potentials and arbitrary electrolytes, i.e., asymmetrical electrolytes. No explicit simple solution was presented for high surface potentials and arbitrary electrolytes.3 For the latter case, an exact solution for ψ becomes very cumbersome. However, we can make use of the concept advanced earlier, namely, that for a high surface potential it is only the counterions that contribute in establishing the potential in the diffuse double layer, and obtain an elegant analytic solution of the Poisson–Boltzmann equation. Let us consider the primitive form of the Poisson–Boltzmann equation, Eq. (5.11):
 N  zi eψ d 2ψ zi eni∞ exp − ǫ 2 =− dx kB T i=1

(5.59)

As we are only interested in the counterions, we can disregard the summation in Eq. (5.59) and write
 zeψ d 2ψ (5.60) ǫ 2 = −zen∞ exp − dx kB T where z and n∞ refer to the counterions. The term − 3

zeψ >0 kB T

The electric potential can be given implicitly by   du κx = −sign(s )  2 k νk [exp(−uzk ) − 1] s

where

νk =

nk∞ ni∞ zi2

i

Here u is an arbitrary function of integration and κ2 =

e2  ni∞ zi2 ǫkB T i

(5.61)

5.2 POTENTIAL FOR PLANAR ELECTRIC DOUBLE LAYER

129

as z and ψ have opposite signs. Consequently, for high surface potentials, one can write 

zeψ zeψ ≫ exp (5.62) exp − kB T kB T Utilizing the inequality presented by Eq. (5.62), one can write for Eq. (5.60)


zeψ exp  d 2ψ k BT ǫ 2 ≃ 2zen∞   dx 




 zeψ − exp − kB T    2

(5.63)

which can be simplified to
 zeψ d 2ψ ǫ 2 = 2zen∞ sinh dx kB T

(5.64)

The above equation is similar to Eq. (5.14) except for the very important fact that z and n∞ refer to the counterions. The solution of Eq. (5.64), subject to the usual boundary conditions ψ(0) = ψs and ψ(∞) = 0, is simply given by  = 2 ln



1 + exp(−κ ∗ x) tanh(s /4) 1 − exp(−κ ∗ x) tanh(s /4)



(5.65)

where s = zeψs /(kB T ) and ∗

κ =




2e2 z2 n∞ ǫkB T

1/2

(5.66)

It should be noted that z is the valence of the counterions and n∞ is the number concentration of the counterions at a location far away from the surface, i.e., bulk solution. Although Eq. (5.65) is similar to that obtained for a (z : z) electrolyte as given by Eq. (5.16), it is different by the manner the Debye length is calculated. EXAMPLE 5.6 Counterion Analysis. Consider the case of a 0.01 M Na2 SO4 solution at 298 K. Compare the electric potential in the diffuse double layer using the counterion solution, Eq. (5.65), with the Debye–Hückel solution for a (1 : 2) electrolyte. Solution

Let us first evaluate κ ∗ given by Eq. (5.66). We can write n∞ = 1000NA M

130

ELECTRIC DOUBLE LAYER

Figure 5.13. Potential distribution near a flat surface. Comparison between the Debye–Hückel and counterion approximation solutions.

and for the system at hand T = 298 K, ǫ = 6.95 × 10−10 C2 /Nm2 , e = 1.602 × 10−19 C, and NA = 6.022 × 1023 mol−1 . Using these values, we obtain √ κ ∗ = 2.32 × 109 z M Let the surface potential be positive, then the counterion valence is z = 2 and M = 0.01. It follows that κ ∗ = 4.64 × 109 m−1

For the case of the Debye–Hückel solution κ = 0.568 × 109 m−1 for a (1 : 2) electrolyte. This value of κ is to be used with the Debye–Hückel approximation, which is given by  = s exp(−κx)

(5.67)

Figure 5.13 shows the variation of the solutions provided by Eqs. (5.65) and (5.67) for s = 2. Both solutions are fairly close to each other. This is to be expected as the Debye–Hückel approximation remains valid for s as high as 2. The latter was shown to be true in the plot of Figure 5.6 where a (z : z) electrolyte was used. For a case of high surface potential in an asymmetrical electrolyte, the solution for  provided by Eqs. (5.65) and (5.66) can be readily used.

5.3 POTENTIAL FOR CURVED ELECTRIC DOUBLE LAYER 5.3.1

Spherical Geometry: Debye–Hückel Approximation

In contrast to the case of planar diffuse double layers, the equations for the diffuse double layers near curved surfaces are more difficult to solve. When a spherical

5.3 POTENTIAL FOR CURVED ELECTRIC DOUBLE LAYER

131

Figure 5.14. Diffuse double layer for a curved surface. Distance from the surface is given by x or (r − a).

surface is considered, the corresponding Poisson–Boltzmann equation in spherical coordinates for a (z : z) electrolyte solution is given as 1 d r 2 dr



  zeψ dψ 2ezn∞ sinh r2 = dr ǫ kB T

(5.68)

The Poisson–Boltzmann equation above assumes the variation of ψ in the radial direction only. The typical boundary conditions are ψ = ψs

ψ →0

at r = a

as r → ∞

(5.69-a) (5.69-b)

Figure 5.14 shows the geometry of the system. In its non-linear form, the Poisson–Boltzmann Eq. (5.68) has no analytical solution subject to the boundary conditions of Eqs. (5.69-a) and (5.69-b). It is analogous to the Gouy–Chapman analysis for a planar surface. In the words of Dukhin (Dukhin and Derjaguin, 1974, p131), “No exact, or even approximate, analytical solution has been found for [Eq. (5.68)] that would be valid for all values of the parameters.” Comments as to the availability of analytical solutions to Eq. (5.68) were also made nearly thirty years after Dukhin’s statement. See for example Wang et al. (2002) and Ohshima (2002). It is more convenient to consider limiting cases for the solution of the Poisson–Boltzmann equation. For thin double layers, i.e., κa ≫ 1, one can rescale Eq. (5.68) with 
X r =a 1+ κa

(5.70)

132

ELECTRIC DOUBLE LAYER

where X = κx and a is a characteristic length, which is the radius in case of a spherical particle. Letting  = zeψ/(kB T ), and making use of Eq. (5.70), the Poisson–Boltzmann equation, Eq. (5.68) becomes 2 d d 2 + = sinh  2 dX κa[1 + X/(κa)] dX

(5.71)

The boundary conditions of Eq. (5.69-a) and (5.69-b) transform to  = s

at X = 0

(5.72-a)

as X −→ ∞

(5.72-b)

and  −→ 0

Note that for κa → ∞ we recover the equations corresponding to a planar diffuse double layer. As it was pointed out by Hunter (1981), it is justifiable to use the planar (flat) surface analysis for a curved surface when the local radius of curvature of a particle, a, is large compared to the double layer thickness, i.e., κa ≫ 1. For example, for particles as small as 0.1 µm, the value of κa is about 10 for a 0.001 M (1 : 1) electrolyte solution. In such cases, the planar analysis of the previous sections is quite applicable to the case of a spherical geometry. Let us now consider the Poisson–Boltzmann equation, Eq. (5.71). Although the solution of Eq. (5.71) requires a numerical route, linearization of the equation simplifies the problem enough to obtain an analytical solution. Letting sinh  ≈  for  ≪ 1 one can obtain a solution to Eq. (5.71) in the form  = s (a/r) exp[−κ(r − a)]

(5.73-a)

 = s (a/r) exp[−κa(r/a − 1)]

(5.73-b)

or

which shows that the decay in  is of an exponential type. Although, expressions for  were derived for s ≪ 1, in practice they are valid for s as high as 2 to 3 for a (1 : 1) electrolyte. The normalized electric potential variation away from the surface is plotted for different values of κa in Figure 5.15. Clearly, for κa = 0.1 and 1, there is much difference between the solution for a sphere and a planar surface. However, for κa = 10 the variation of the normalized potential with κx differs little from the Debye–Hückel solution for a planar surface. This is in accordance with the statements made by Hunter (1981) regarding the use of planar surface solutions for κa ≥ 10. Loeb et al. (1961) give a comprehensive numerical treatment of the electric potential variation for a spherical geometry for both symmetric and asymmetric electrolytes.

5.3 POTENTIAL FOR CURVED ELECTRIC DOUBLE LAYER

133

Figure 5.15. Diffuse double layer for planar and spherical surfaces.

EXAMPLE 5.7 Debye–Hückel Analysis. It is of interest to compare Debye–Hückel solutions for planar and curved surfaces. Let us consider a 0.001 M solution of a (2 : 1) electrolyte. Let the particle radius, a, vary from 2 to 100 nm. Compare the variation for ψ with distance away from a flat and a spherical surface. Solution The Debye–Hückel approximation for a planar surface is given by  = s exp(−κx)

(5.74)

Recast the Debye–Hückel solution for a spherical surface using x = r − a. Here, x denotes distance from the surface. Eq. (5.73-a) gives
 1  = s exp[−κx] (5.75) 1 + κx/κa The difference between the two geometries is through the term 1/(1 + κx/κa) which is only important for small values of the radius of curvature, a, and at intermediate proximity to the surface. Plots of Eqs. (5.74) and (5.75) are shown in Figure 5.16. For the smallest particle radius, a = 2 nm, the curvature effect is significant as there is large departure from the planar surface. For a = 100 nm, there is little difference in  distribution using both expressions. Here, the curvature effects are negligible. The surface charge density for a charged sphere can be derived as follows. Due to electroneutrality, the total surface charge on the sphere surface is balanced by the charge in the electric double layer. Therefore,  ∞ Qs = − 4π r 2 ρf dr a

134

ELECTRIC DOUBLE LAYER

Figure 5.16. Diffuse double layer for a sphere and a planar surface.

and the surface charge density is given by qs =

Qs 1 =− 2 4π a 2 a



∞

r 2 ρf dr

(5.76)

a

where r is the radial component of the spherical coordinate system and a is the sphere’s radius. Poisson’s equation in spherical coordinates is given by 1 d r 2 dr




r

2 dψ

dr



=−

ρf ǫ

(5.77)

Substituting for ρf using Eq. (5.77) and simplifying Eq. (5.76) gives qs =

ǫ a2



∞

a


 dψ d r2 dr

(5.78)

Upon integration, Eq. (5.78) gives ǫ qs = 2 a Recognizing that r2 Eq. (5.79) gives



     2 dψ  −r r  dr ∞ dr a 2 dψ 

 dψ  →0 dr ∞

(5.79)

as r → ∞

 dψ  qs = −ǫ dr a

(5.80)

5.3 POTENTIAL FOR CURVED ELECTRIC DOUBLE LAYER

135

The relationship between the surface charge density given by Eq. (5.80) is a consequence of the Gauss theorem and it is similar to that of a planar surface. Making use of the electric potential expression, Eq. (5.73-a), Eq. (5.80) gives
 1 (5.81) qs = ǫκψs 1 + κa Equation (5.81) relates the surface charge density to the surface potential for a charged spherical geometry. In the limit of a very small Debye length, i.e., κa → ∞, Eq. (5.81) reduces to the case of a planar surface, where qs = ǫκψs

(5.82)

The relationship between the surface charge density, qs , and the surface potential given by Eq. (5.81) is strictly valid for low surface potentials. A more general equation, albeit approximate, for a symmetric electrolyte is given by Loeb et al. (1961) as   ǫκkB T 4 qs = tanh(s /4) (5.83) 2 sinh(s /2) + ez κa Ohshima et al. (1982) provided justification for the use of Eq. (5.83). However, Lyklema (1995) pointed out that the potential distribution derived from Eq. (5.83) does not yield a very good approximation. For s ≪ 1, Eq. (5.83) reduces to
 1 ǫkB T κ (5.84-a) qs = 1+ s ez κa or
 1 qs = ǫκψs 1 + κa

(5.84-b)

qs = ǫψs /a

(5.85-a)

which is identical to the previously derived expression given by Eq. (5.81). Now let us consider Eq. (5.81) for the case of κ → 0. Such a case is applicable for very low electrolyte concentrations and it is particularly true for non-aqueous systems where the ionic concentration is usually very low in the continuous medium. For κ → 0, Eq. (5.81) reduces to

In terms of total surface charge, the equivalent expression for the case of κ → 0 is Qs = 4π aǫψs

(5.85-b)

The above expression is identical to the case of a charged sphere in a dielectric medium. It was derived using electrostatic procedures, see Example 3, Chapter 3.

136

ELECTRIC DOUBLE LAYER

For very large electric double layers, κ −1 → ∞, expressions for surface potential and charge, derived using electrostatics arguments for a charged body in a dielectric, are useful in electrokinetics studies. White (1977), Ohshima et al. (1982), and Ohshima (1995) gave improved solutions to the potential distribution for a charged spherical particle. Ohshima (2002) gave approximate expressions for the surface charge density – surface potential relationship for a spherical colloidal particle in a salt-free aqueous medium, as well as, for non-aqueous media containing only counterions. He showed the suitability of using the relationship given by Eq. (5.85-b) for the case of low potentials and large Debye lengths. 5.3.2

Cylindrical Geometry: Debye–Hückel Approximation

For a cylindrical geometry, there exists no closed form analytical solution of the Poisson–Boltzmann equation and the comments pertaining to a spherical geometry are equally valid for the case of a cylindrical geometry. The Poisson–Boltzmann equation in cylindrical coordinates (r-component only) for a (z : z) electrolyte is given by

  zeψ dψ 2ezn∞ 1 d sinh r = (5.86) r dr dr ǫ kB T For low potential, it reduces to 1 d r dr


 dψ r = κ 2ψ dr

(5.87)

Here, r is the radial coordinate. For the boundary conditions of ψ(0) = ψs and dψ(∞)/dr = 0, the solution of Eq. (5.87) was given by Dube (1943) (see Lyklema, 1995) as ψ = ψs

K0 (κr) K0 (κa)

(5.88)

where K0 is the zeroth-order modified Bessel function. The normalized potential distribution for the case of a cylinder is shown in Figure 5.17. As in the case of a spherical geometry, large deviation from the planar type distribution is present for small values of κa. However, for κa > 10, the potential distribution is fairly close to the Debye–Hückel solution for a planar surface. It is of interest to compare the potential distribution for the cases of planar, cylindrical, and spherical geometries. Following Lyklema’s (1995) analysis, we can write the Poisson–Boltzmann equation in the form
 d p dψ + = κ 2ψ (5.89) dr r dr where r is the distance coordinate normal to a surface. The cases of p = 0, 1, and 2 correspond to planar, cylindrical, and spherical geometries, respectively. The values

5.3 POTENTIAL FOR CURVED ELECTRIC DOUBLE LAYER

137

Figure 5.17. Diffuse double layer for a cylindrical surface.

of p would suggest that a spherical geometry would appear as “more curved” for a given κa value and the potential distribution for a cylindrical geometry would lie between the spherical and planar geometries. Figure 5.18 shows the normalized potential distribution for κa = 1 and 10 for planar, cylindrical, and spherical geometries. Clearly, the ψ/ψs distribution for the cylindrical geometry lies between those of the spherical and planar geometries as suggested by Lyklema (1995). All geometries considered here give the same ψ/ψs distribution for large values of κa, say κa > 50. The extension of the potential distribution outlined above for the case of a cylindrical geometry is provided by Ohshima (1998). The solutions of the Poisson–Boltzmann equation considered here were limited to simple geometries of an infinite planar

Figure 5.18. Diffuse double layer for planar, cylindrical, and spherical surfaces.

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ELECTRIC DOUBLE LAYER

surface, long cylinders, and spheres. In essence, the solutions were provided for one-dimensional problems. For geometries requiring solution of the two-dimensional Poisson–Boltzmann equation, various analyses are available in the literature. Solutions were provided for the case of spheroids by Fair and Anderson (1989), Feng and Wu (1994), Keh and Huang (1993), and Hsu and Liu (1996). In the case of a spheroidal geometry, a rod shaped colloidal particle can be approximated by a prolate spheroid and a red blood cell shape by an oblate spheroid. Electric double layer interaction between a spherical particle and a cylinder was provided by Gu (2000). 5.4 ELECTROSTATIC INTERACTION BETWEEN TWO PLANAR SURFACES When two charged surfaces (or two colloidal particles), each surrounded by an electrical diffuse double layer, approach one another, their respective double layers begin to overlap. As a result, the ionic and potential distributions around a given particle, when it is brought in the vicinity of a second particle, are no longer symmetrical. This causes asymmetrical stresses of electrical origin on the particle surface and, as a result, the particle experiences a force. The evaluation of the force can be made from the solution of the Poisson–Boltzmann equation for the potential coupled with the momentum conservation equation. For the sake of simplicity (and since analytical solutions of the relevant equations can be obtained readily), we shall first consider computation of the force per unit area between two charged infinite flat plates in a dielectric medium containing free charge. 5.4.1

Force between Two Charged Planar Surfaces

At the outset, it should be stated that no real system conforms to the picture of two interacting infinite parallel plates. This geometry is a highly simplified approximation of typical colloidal systems based on scaling arguments. Generally, colloidal particles are bodies with curvature (for instance spheres) suspended in a large volume of electrolyte solution. The forces that we are interested in are caused by overlap of the electric double layers when two such colloidal particles approach each other. Typically, the gaps between the particles at which overlap between their double layers becomes perceptible are of the order of nanometers. On the other hand, radii of curvature of the colloidal particles are often of the order of micrometers. Consequently, the gaps at which these forces become measurable are negligible compared to the radii of the interacting particles. In these cases, it is often possible to completely ignore the curvature of the particles, and treat them as infinite planar surfaces. As a consequence of this approximation, however, throughout the subsequent analysis, it should be borne in mind that although we are considering interaction between two infinite planar surfaces, the surfaces actually belong to finite bodies suspended in a large bath of electrolyte solution. Thus, the electrolyte solution trapped between these planar surfaces is connected to a reservoir of electroneutral electrolyte, in which, at sufficiently large distances from the charged surfaces, the ion concentrations take their bulk values, and the electric potential becomes zero.

5.4 ELECTROSTATIC INTERACTION BETWEEN TWO PLANAR SURFACES

139

Figure 5.19. Overlap of two planar double layers.

Consider two flat plates at constant surface potentials of ψa and ψb as shown in Figure 5.19. These surface potentials will be assumed to remain constant irrespective of the separation between the plates. The Poisson equation for the case of parallel plates is given by d 2ψ ǫ 2 = −ρf (5.90) dx which, with the Debye–Hückel approximation, becomes d 2ψ = κ 2ψ dx 2

(5.91)

The momentum equation governing liquid flow, as given by the Navier–Stokes equation including the appropriate electric body force is given by µ∇ 2 u + ρg + ρf E = ∇p

(5.92)

where u is the liquid velocity vector, µ is the liquid viscosity, p is the pressure, g is the gravitational acceleration, and ρ is the liquid mass density. The term ρf E represents the electric body force per unit volume, comprising of the free charge density, ρf , and the local electric field E. We will discuss the Navier–Stokes equation in greater detail in Chapter 6. For a stationary system, since there is no fluid flow, we have u = 0. Further, neglecting the effect of gravity, we can write Eq (5.92) as ∇p = ρf E

(5.93)

For a one-dimensional problem (when the only space variable is x), the above equation simplifies to dp = ρf Ex (5.94) dx

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ELECTRIC DOUBLE LAYER

where dψ dx One can combine Eqs. (5.94) and (5.95) to obtain

(5.95)

Ex = −

dp dψ + ρf =0 dx dx

(5.96)

Combining Eqs. (5.96) with the Poisson equation (5.90) leads to d 2 ψ dψ dp −ǫ 2 =0 dx dx dx

(5.97)

The above equation relates the pressure to the electric potential ψ. The pressure arises due to variation in the ionic concentration between the plates and the surrounding electrolyte solution. Integration of Eq. (5.97) gives ǫ p− 2




dψ dx

2

(5.98)

= C1

where C1 is the integration constant and is the required force per unit area to keep the two plates from moving. For this particular case, it turns out that the net force per unit area Fp /Ap , which is the difference between the pressure p and the electric force per unit area, is constant everywhere within two plates.4 Replacing C1 by Fp /Ap , Eq. (5.98) becomes
 Fp ǫ dψ 2 (5.99) =p− Ap 2 dx We need to determine p and ψ in order to evaluate the force per unit area, Fp /Ap . The solution of Eq. (5.91) gives ψ = A1 cosh κx + B1 sinh κx.

(5.100)

Making use of the boundary conditions of ψ at x = ±h/2, where ψ(h/2) = ψa and ψ(−h/2) = ψb , the constants of integration are given as A1 =

ψa + ψ b 2 cosh(κh/2)

and B1 =

ψa − ψb 2 sinh(κh/2)

(5.101)

For the special case of ψa = ψb , B1 = 0. The pressure, p, in Eq. (5.99) can be derived from Eq. (5.93). Substituting the expression for the free charge density, ρf , in Eq. (5.93) we can write  
 zi eψ ∇ψ = 0 (5.102) ∇p + e zi ni∞ exp − kB T i 4

For a derivation of the force per unit area, refer to Example 5.8.

5.4 ELECTROSTATIC INTERACTION BETWEEN TWO PLANAR SURFACES

141

where ni∞ is the bulk concentration of the i th ionic species. Equation (5.102) can be rewritten as 
  zi eψ ∇ p − kB T ni∞ exp − =0 (5.103) kB T i

Since the term in the square brackets in Eq. (5.103) is independent of the coordinates, it is a constant. Consequently, one can equate the term at any point between the plates, where the electric potential has a finite value, with another point in the bulk electrolyte solution, where ψ = 0 and p = p∞ = 0.5 In other words, 
  zi eψ ni∞ (5.104) = −kB T p − kB T ni∞ exp − kB T i i Rearrangement of Eq. (5.104) yields p = kB T

 i

  
zi eψ −1 ni∞ exp − kB T

(5.105)

For low potentials, zi eψ/kB T ≪ 1, and a symmetric electrolyte, Eq. (5.105) simplifies to
 zeψ 2 p = n ∞ kB T (5.106) kB T

It may be noted here that in literature, the pressure term is often attributed to the osmotic pressure. This is perhaps owing to the form of the right hand side term in Eq. (5.105), which is essentially kB T i (ni − ni∞ ). This expression is identical to the Van’t Hoff formula for osmotic pressure. From a thermodynamic viewpoint, this expression for the osmotic pressure can be derived from the Gibbs–Duhem equation as well. However, it is clear from the above derivation that the pressure is of a hydrostatic origin, and can be derived in a straightforward manner from the Navier– Stokes equation considering the electric body force. In this context, one may consider Eq. (5.93) as an equivalent form of the Gibbs–Duhem equation. Substitution of the expression for p, Eq. (5.106), in the expression for the force per unit area, Eq. (5.99), yields Fp = n∞ kB T Ap




zeψ kB T

2

ǫ − 2




dψ dx

2

One can now use the equation for ψ, Eq. (5.100), in the above expression to obtain
2 2   Fp e z n∞  2 = (5.107) A1 − B12 Ap kB T 5

In assuming p∞ = 0 we are simply writing the hydrostatic pressure in terms of gauge pressure with the pressure in the bulk electrolyte as the reference.

142

ELECTRIC DOUBLE LAYER

Using the expressions for A1 and B1 from Eq. (5.101) in Eq. (5.107) yields Fp e 2 z 2 n∞ = Ap kB T



2ψa ψb cosh(κh) − ψa2 − ψb2 sinh2 (κh)



(5.108)

In terms of Debye length, the force per unit area is given as Fp ǫκ 2 = Ap 2



2ψa ψb cosh(κh) − ψa2 − ψb2 sinh2 (κh)



(5.109)

where κ2 =

2e2 z2 n∞ ǫkB T

We note here that Eqs. (5.108) and (5.109) are identical, and comparing the pre-factors (the terms on the right hand sides outside the square brackets) in these equations, we obtain ǫκ 2 e 2 z 2 n∞ = kB T 2 Rearranging the above equation yields n∞ kB T ǫ = κ2 2




kB T ze

2

which is the same relationship as Eq. (5.56). For the special case of ψa = ψb = ψs , Eqs. (5.108) and (5.109) reduce to Fp = Ap




2e2 z2 n∞ ψs2 kB T



cosh(κh) − 1 sinh2 (κh)



(5.110)

Simplification of Eq. (5.110) leads to Fp ǫκ 2 ψs2 = Ap [cosh(κh) + 1]

(5.111-a)

Fp ǫκ 2 ψs2 = Ap 2 cosh2 (κh/2)

(5.111-b)

or

Either Eq. (5.108) or Eq. (5.109) gives the force per unit area for the general case of two planar charged surfaces having different surface electric potentials. Equations (5.111-a) and (5.111-b) give the force per unit area for the special case of surfaces having the same surface potential.

5.4 ELECTROSTATIC INTERACTION BETWEEN TWO PLANAR SURFACES

143

Let us consider the limiting cases of κh for surfaces having the same potentials (i) Very small κh: Making use of the expansion for cosh2 (κh/2), Eq. (5.111-b) gives Fp ǫκ 2 ψs2 [1 − (κh)2 /4] = (5.112) Ap 2 Equation (5.112) indicates that for small κh, the force per unit area is proportional to ψs2 and is weakly dependent on the separation distance between the two plates. (ii) Very large κh: In this limit, namely, κh ≫ 1, h → ∞, Eq. (5.111-a) gives Fp = 2ǫκ 2 ψs2 exp(−κh) Ap

(5.113)

As would be expected, the interaction force is zero at very large separation distances. Additional details may be found in Russel et al. (1989), Probstein (2003), van de Ven (1989), Ross and Morrison (1988), and Chan et al. (1975, 1976). For a given Debye length, when the two surfaces approach each other, the electric double layers associated with each surface will interact with each other. Figure 5.20 shows the normalized potential distribution for two similar surfaces for κ = 1.0 nm−1 . For h = 50 nm, the potential at the mid-plane is zero and for h = 10 nm it is nearly zero. However, when h < 10 nm the mid-plane normalized potential, ψ(0)/ψs , is not zero and here the two electric double layers interact. Such an interaction is very important. Surfaces with dissimilar potentials but of the same sign can give rather unexpected results. At small separations, it is possible to have an attractive force between two such

Figure 5.20. Potential distribution between two planar surfaces at various separation distances.

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ELECTRIC DOUBLE LAYER

Figure 5.21. Electrostatic double layer force between two constant potential surfaces. Case (a): a = 0.5 and b = 1.0; Case (b): a = b = 0.5.

surfaces. This behavior at close separations is due to a change in the sign of the surface charge on one of the surfaces due to the overlapping double layers, while the surface potentials are assumed to remain constant. The very fact that surfaces with dissimilar potentials but of the same sign can encounter an attractive force on close approach has a major impact in the area of hetero-coagulation, Russel et al. (1989). Figure 5.21 shows the variation of the dimensionless force per unit area with separation distance, κh, for two cases. In the case of a = 0.5 and b = 1.0, the electrostatic force becomes attractive at a dimensionless separation distance of less than κh = 0.693. However, for the case of a = b = 0.5, the force remains repulsive at all separation distances. In the next section we will discuss overlapping double layers in more detail. EXAMPLE 5.8 Force between Charged Parallel Plates. Derive an expression for the force per unit area between two parallel capacitor plates having surface potentials ψA and ψB separated by a distance 2h in a symmetric (z : z) electrolyte bath with bulk concentration n∞ . Solution We consider two thin charged plates A and B with constant surface potentials ψA and ψB , respectively, immersed in a large electrolyte bath as shown in Figure 5.22. The electric potential and the ion concentration of the bath far away from the plates are zero and n∞ , respectively. The pressure in this bulk electrolyte region is denoted by p∞ . The origin of the Cartesian coordinate system is at the midplane of the two parallel plates. The plate dimensions along the y and z coordinate directions are considered to be much larger than the gap between them along the x coordinate direction. To compute the force between the plates, it is sufficient to determine the total force acting on one of the plates. This force can be determined by integrating the total

5.4 ELECTROSTATIC INTERACTION BETWEEN TWO PLANAR SURFACES

145

Figure 5.22. Schematic diagram for the calculation of the interaction force per unit area between two charged parallel capacitor plates.

stress tensor over the entire surface of one of the plates. Focussing on the plate B, = let us consider the different stresses acting on it. The total stress, T , will have two components. First, we have the static pressure of the fluid, −p, acting normal to the surface of the plate at every point, the negative sign implying that the pressure is directed toward the plate (opposite to the direction of the unit surface normal, n). = This stress, Tp , is explicitly written as   −p 0 0 = = 0 Tp = −p I =  0 −p 0 0 −p =

where I is the unit tensor. The second component is due to the electrical (Maxwell) = stress, Te . The electrical stress is given by the stress tensor, Eq. (3.156), = = 1 Te = ǫEE − ǫE · EI 2

where E is the electric field vector, and ǫ is the dielectric constant of the fluid (electrolyte solution). Making use of the assumption that the plate dimensions along y and z are much larger than the gap between them, the electric field can be assumed to be one dimensional, i.e., E = Ex ix , where ix is a unit vector along the positive x coordinate. Consequently, from Eq. (3.162), the electrical stress tensor becomes  2  0 0 ǫEx /2 = 0  −ǫEx2 /2 Te =  0 0 0 −ǫEx2 /2 =

The total stress, T , is given by  2 ǫEx /2 − p = = = 0 T = Tp + Te =  0

0 −ǫEx2 /2 − p 0

 0  0 −ǫEx2 /2 − p

(5.114)

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ELECTRIC DOUBLE LAYER

The total force on plate B is obtained by integrating the total stress over its entire surface. In other words  =

T · n dS

Fp =

S

where S is the total surface area of the plate. Note that this integration is similar to = Eq. (3.161), except that the pressure is also included in the total stress tensor, T . Neglecting the contribution to the force arising from the extremities of the plate, one can observe that the total force can be determined by integrating the stress tensor over the plate cross sectional area Ap on the inside and outside faces of plate B (denoted by “in” and “out”, respectively). Furthermore, since the stress does not depend on the y and z coordinates, we obtain Fp =



in

=

T · n dAp +



=

=

=

T · n dAp = [T · n]in Ap + [T · n]out Ap

(5.115)

out

On the inside face of the plate B, n = −ix , while on the outside face, n = ix . Thus, = = Fp = −[T · ix ]in + [T · ix ]out Ap

Writing the dot product of the total stress tensor with the unit vector ix explicitly yields
2  
2 Fp ǫEx ǫEx =− −p −p + Ap 2 2 in out

 2 The quantity ǫEx /2 − p out , evaluated on the outside surface of the plate, is a constant at every location x. Now, since p → p∞ and Ex → 0 at a large distance from the plate, x → ∞ (in the bulk electrolyte solution), one can write


ǫEx2 −p 2



out

= −p∞

Therefore

or


2  Fp ǫEx =− −p − p∞ Ap 2 in

(5.116)


2 Fp E = pin − p∞ − ǫ x Ap 2 in

(5.117)

The above expression for the force per unit area is a slightly more generalized form of Eq. (5.99). The methodology described above gives the force on plate B. The force on plate A will be equal and opposite to the force on plate B. The force on B is repulsive if it acts along ix (positive x direction), and attractive if it acts along the negative x direction.

5.4 ELECTROSTATIC INTERACTION BETWEEN TWO PLANAR SURFACES

147

5.4.2 Surface Charge Density for Planar Surfaces: Overlapping Double Layers Similar to the analysis conducted for an isolated charged surface, we can proceed in evaluating the surface charge densities as a function of the surface potentials and separation distance. We will carry out the analysis for similar surfaces using the solution for the potential distribution obtained in the previous section under the assumption of constant surface potentials. For convenience, however, in the case of dissimilar surfaces, we will utilize the potential distribution solution under the assumption of constant surface charge densities. In the following analysis, the geometry depicted in Figure 5.19 is used. 5.4.2.1 Similar Surfaces Due to symmetry, for the case of similar surfaces only one side of the mid-plane needs to be considered. To conserve electroneutrality, the charges in the electric double layers and on the surface must be conserved. This leads to  h/2

qs = −

0

ρf dx

for the right hand side half-width. Making use of the Poisson equation, we can write   h/2 2  h/2
d ψ dψ qs = ǫ d dx = ǫ dx 2 dx 0 0

(5.118)

Upon integration, we obtain qs = ǫ



   dψ  dψ  − dx h/2 dx 0

Due to symmetry, at the mid-point between the two surfaces, (5.119) becomes  dψ  qs = ǫ dx h/2

(5.119) 

dψ  dx 0

= 0 and Eq. (5.120)

or

qs = −ǫEs

(5.121)

where Es is the electric field at the surface. To evaluate qs we need to use an expression for the potential distribution. To that end, when the surface potentials on both the surfaces are equal, Eq. (5.100) gives ψ=

ψs cosh(κx) cosh(κh/2)

(5.122)

Making use of Eq. (5.122), the surface charge distribution of Eq. (5.120) becomes qs = ǫκψs tanh(κh/2)

(5.123)

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ELECTRIC DOUBLE LAYER

Equation (5.123) relates the surface charge density to the surface potential and the separation gap, h, between the two similar surfaces. For a constant surface potential ψs and given ǫ and κ, the surface charge density becomes a function of the separation gap, h. In other words, the surface charge density, qs , is not a constant but a function of the separation distance between the two charged surfaces whose surface potentials are kept constant for all values of h. As tanh(κh/2) is positive for all values of κh/2, the surface charge density, qs , does not change signs with changes in the gap width h or κh. As tanh(0) = 0 and tanh(∞) = 1, the dimensionless surface charge density (qs /ǫκψs ) varies from zero corresponding to zero gap to unity at large separations. The force between two planar similar surfaces can be given in terms of the surface charge density qs rather than the surface potential, ψs . Combining Eqs. (5.110) and (5.123) leads to Fp q2 [cosh(κh) − 1] = s Ap ǫ sinh2 (κh) tanh2 (κh/2) Simplifying, one obtains Fp q2 1 = s Ap ǫ [cosh(κh) − 1]

(5.124-a)

Fp q2 1 = s Ap ǫ 2 sinh2 (κh/2)

(5.124-b)

or

For κh ≪ 1, Eq. (5.124-b) gives Fp 2q 2 = 2s 2 Ap ǫκ h

(5.125)

and for κh ≫ 1, Eq. (5.124-a) gives Fp 2q 2 exp(−κh) = s Ap ǫ

(5.126)

For the case of κh ≫ 1, by holding either ψs or qs constant the force per unit area approaches zero at large separation distances as indicated by Eqs. (5.113) and (5.126). For the case of κh ≪ 1, i.e., at very small separation distances, holding the surface potential constant gives a force per unit area that is weakly dependent on the separation gap as indicated by Eq. (5.112). However, holding the surface charge density constant gives a very large repulsive force at small separations as indicated by Eq. (5.125). It is accepted in literature that at small separations between two interacting surfaces, it is more likely that the surface potential holds constant rather than the surface charge remains constant. In real systems, it is likely that neither the surface charge nor the surface potential remains constant on close approach of the two surfaces. Charge regulation takes place on close approach and it plays an important role in flocculation and colloidal particle attachment to surfaces.

5.4 ELECTROSTATIC INTERACTION BETWEEN TWO PLANAR SURFACES

149

5.4.2.2 Dissimilar Surfaces The potential distribution for two charged planar surfaces is given by ψ = A2 cosh(κx) + B2 sinh(κx)

(5.127)

The coefficients A2 and B2 can be evaluated using the boundary conditions of constant surface charge density where at x = h/2, qs = qa and at x = −h/2, qs = qb as shown in Figure 5.19. The constants are provided by Russel et al. (1989) as A2 =

(qa + qb ) 2ǫκ sinh(κh/2)

(5.128)

B2 =

(qa − qb ) 2ǫκ cosh(κh/2)

(5.129)

and

Let us make use of the solution for the potential distribution given by Eqs. (5.127)– (5.129) to enable us to discuss surface charge reversal observed during interaction between dissimilar planar surfaces at close approach. The surface potentials at x = ±h/2, i.e., ψ(h/2) = ψa and ψ(−h/2) = ψb can be given by Eqs. (5.127)–(5.129) as ψa =

(qa − qb ) tanh(κh/2) (qa + qb ) + 2ǫκ tanh(κh/2) 2ǫκ

(5.130)

ψb =

(qa + qb ) (qa − qb ) tanh(κh/2) − 2ǫκ tanh(κh/2) 2ǫκ

(5.131)

and

Solving for qa and qb , Eqs. (5.130) and (5.131) give   (ψa − ψb ) ǫκ (ψa + ψb ) tanh(κh/2) + qa = 2 tanh(κh/2)

(5.132)

  ǫκ (ψa − ψb ) (ψa + ψb ) tanh(κh/2) − 2 tanh(κh/2)

(5.133)

and qb =

Clearly, Eqs. (5.132) and (5.133) show that by keeping the surface potentials constant, the surface charge densities vary with the dimensionless separation gap between the plates. Let us consider the case of dissimilar surfaces held at constant potentials where ψa = ψb and both ψa and ψb have the same sign. Without loss of generality, let us also assume that ψa < ψb and that both ψa and ψb are positive. Consideration of Eq. (5.133) would indicate that the surface charge density, qb , will not change signs for all values of the dimensionless separation gap, κh. However, the surface charge

150

ELECTRIC DOUBLE LAYER

density, qa , can change signs. The dimensionless gap width at which qa changes signs can be evaluated by setting qa = 0 in Eq. (5.133). This leads to 
ψa − ψb tanh2 (κh/2) = − (5.134) ψa + ψb The variations of qa and qb for the cases of a = b = 0.5 and a = 0.5 and b = 1.0 are shown in Figure 5.23. For the case of similar surfaces, i.e., a = b = 0.5, both qa and qb remain positive for all values of κh. For the case of dissimilar surfaces (a < b ), qb is positive for all κh values. However, the surface having the lower surface potential experiences a surface charge reversal that will eventually lead to an attractive force between the two dissimilar surfaces. The charge reversal for a = 0.5 and b = 1.0 according to Eq. (5.134) occurs at  κh = 2 tanh−1 1/3 = 1.317

The location of the surface charge reversal is shown on Figure 5.23. As the surface charge density is proportional to the gradient of the electrical potential at the surface, a surface charge reversal should occur when n · ∇ψ = 0

(5.135-a)

or  dψ  =0 dn surface

(5.135-b)

Figure 5.23. Surface charge regulation for two planar surfaces with dimensionless separation distance.

5.4 ELECTROSTATIC INTERACTION BETWEEN TWO PLANAR SURFACES

151

A plot of the electric potential in the gap between two parallel plates is shown in Figure 5.24 for a = 0.5 and b = 1.0. Clearly at κh = 1.317, the electric potential gradient is zero at x/ h = 0.5. The slope is positive for κh > 1.317 and negative for κh < 1.317. A quick review of Figures 5.21 and 5.23 would indicate that the gap location at which the charge reversal takes place does not coincide with the force reversal from repulsive to attractive. The condition for force reversal from repulsive to attractive can be obtained by setting the force between the two dissimilar plates to zero. Setting Fp /Ap in Eq. (5.107) to zero, one obtains A1 = ∓B1 For a < b one obtains


 a − b tanh(κh/2) = − a + b



or tanh(κh/2) =

b /a − 1 b /a + 1

(5.136)

The dimensionless gap width, κh, at which the force reverses from repulsive to attractive is dependent solely on the ratio of b /a , Eq. (5.136), and it is not the same gap width at which the charge reversal occurs as given by Eq. (5.134). In fact, the repulsive force is maximum at the distance where the charge reverses its sign.

Figure 5.24. Electric potential distribution between two planar surfaces at various separation distances.

152

ELECTRIC DOUBLE LAYER

5.5 ELECTROSTATIC POTENTIAL ENERGY The interaction between particles due to electrostatic potential forces is most conveniently characterized in terms of the electrostatic potential energy, φ. Recognizing that work or energy is related to force by Force = −

d(energy) d(work) =− d(distance) d(distance)

one can write 

h

ǫκ 2 ψs2 2



φ=−

(5.137)

(Force)dx

∞

where, by convention, the potential energy is given by bringing a particle from infinity to a separation distance h. For the case of two parallel plates, using Eq. (5.111-b), the interaction energy per unit area, φp∗ (= φp /Ap ), between the two charged parallel plates is given by φp∗ = −

h

∞

dx cosh2 (κx/2)

leading to φp∗ = ǫκψs2 [1 − tanh(κh/2)]

or

φp∗

=

2ǫκψs2



exp(−κh) 1 + exp(−κh)



(5.138-a) (5.138-b)

For κh ≪ 1, Eq. (5.138-a) provides φp∗ = ǫκψs2 (1 − κh/2)

(5.139-a)

and for κh ≫ 1, Eq. (5.138-b) provides φp∗ = 2ǫκψs2 exp(−κh)

(5.139-b)

EXAMPLE 5.9 Electrostatic Potential Energy. Derive the electrostatic potential energy between two identical spheres carrying the same electric surface potential. The electrostatic force between two charged spheres can be given by F = 2π ǫκaψs2

exp(−κh) 1 + exp(−κh)

(5.140)

where a is the sphere radius and ψs is their surface electric potential. Here h is the separation gap between the two spheres.

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5.5 ELECTROSTATIC POTENTIAL ENERGY

Solution The electrostatic potential energy is given by combining Eqs. (5.137) and (5.140) for the case of κa ≫ 1 φ=−



h

∞

2π ǫκaψs2

exp(−κh)dh 1 + exp(−κh)

(5.141)

Hence,   φ = 2π ǫaψs2 ln 1 + exp(−κh)

(5.142)

The above expression is the electrostatic interaction energy between two spheres having the same radii and surface potentials. From a thermodynamic standpoint, the interaction energy between two double layers at a distance h is defined as the Gibbs free energy change6 associated with bringing the two double layers from an infinite separation to a finite distance φ = G = G(h) − G(∞)

(5.143)

Here, G represents the Gibbs free energy of the entire system comprised of two electric double layers. For each double layer in isolation, the charging process yields a certain free energy, which is represented by (Overbeek, 1990) G=−



ψ0

σ dψ

(5.144)

0

where σ is the surface charge density and ψ is the surface potential. The integral refers to the free energy change associated with charging up a double layer from zero potential (uncharged surface) to a potential of ψ0 . Knowing a relationship between the surface charge density and the surface potential, one can evaluate the integral in Eq. (5.144) and obtain the free energy associated with an isolated electric double layer maintained at a given surface potential ψ0 . For example, considering a planar double layer with a small surface potential, such that the Debye–Hückel approximation holds, we can write, cf., Eq. (5.54), σ = ǫκψ

(5.145)

ǫκψ02 σ0 ψ0 G =− =− Ap 2 2

(5.146)

Substituting Eq. (5.145) in Eq. (5.144) and evaluating the integral one obtains

where σ0 and ψ0 are the surface charge density and surface potential of the isolated planar double layer, and the term G/Ap is used to emphasize that we are dealing with the free energy per unit area for a planar geometry. 6

Helmholtz free energy can be used instead of Gibbs free energy if the pressure volume work is zero.

154

ELECTRIC DOUBLE LAYER

EXAMPLE 5.10 Electric Potential Energy. Consider the case of two planar double layers with constant surface potentials ψa and ψb brought together to a separation distance h from infinity. Derive an expression for the interaction potential energy per unit area between the two double layers at a separation distance h. We will base the analysis on the linearized Poisson–Boltzmann equation, namely, the Debye–Hückel approximation. Solution The geometry under consideration is given in Figure 5.25. Note that the coordinate system used in this example is slightly different from the one used in Figure 5.19. The location of the origin is shifted in this example to ensure consistency with the original derivations of Hogg et al. (1966), which would lead to the expressions obtained here. The solution of the linearized Poisson–Boltzmann equation for this geometry will be of the form, cf., Eq. (5.100), ψ = A1 cosh(κx) + B1 sinh(κx) where the constants A1 and B1 can be obtained from the boundary conditions. The boundary conditions are ψ = ψa at x = 0 and ψ = ψb at x = h. Applying these conditions, the expression for the electrical potential distribution becomes ψ = ψa cosh(κx) +




 ψb − ψa cosh(κh) sinh(κx) sinh(κh)

(5.147)

The surface charge density at the two plates are obtained from the derivative of Eq. (5.147) evaluated at these surfaces. For the plate at x = 0, we have σa = −ǫ




dψ dx



= −ǫκ[ψb cosech(κh) − ψa coth(κh)] x=0

Figure 5.25. Calculation of electrostatic interaction potential between two planar surfaces separated by a distance h.

5.6 ELECTROSTATIC INTERACTIONS BETWEEN CURVED GEOMETRIES

155

while for the plate at x = h we have
 dψ σb = −ǫ = ǫκ[ψb coth(κh) − ψa cosech(κh)] dx x=h To evaluate the interaction potential energy using Eq. (5.143), we need to determine the free energies of the system at a separation h and at infinite separation. The free energy per unit area of the system consisting of two planar double layers is given by adding the free energies per unit area of the individual double layers, Eq. (5.146). Utilizing the expressions for the electric potential and the surface charge density obtained above, one can write, G(h) 1 ǫκ = − (σa ψa + σb ψb ) = [2ψa ψb cosech(κh) − (ψa2 + ψb2 ) coth(κh)] Ap 2 2 and

1 ǫκ G(∞) = − (σa ψa + σb ψb ) = − (ψa2 + ψb2 ) Ap 2 2

Now, from Eq. (5.143), we obtain φ(h) G(h) G(∞) = − Ap Ap Ap ǫκ = {(ψa2 + ψb2 )[1 − coth(κh)] + 2ψa ψb cosech(κh)} 2

(5.148)

Equation (5.148) is the well known HHF (Hogg–Healy–Fuerstenau) (Hogg et al., 1966) expression for the interaction potential energy per unit area between two infinite planar surfaces. This expression applies to constant potential surfaces and is valid in the Debye–Hückel limit of low surface potentials. Integration of the interaction force, Eq. (5.109), would also yield Eq. (5.148), thus emphasizing that the approach based on the evaluation of the interaction force from integration of the stresses and the approach based on the evaluation of the free energy changes are identical. An interesting feature of this result is that the interaction energy remains finite even when one of the surfaces is assigned a zero potential. A similar derivation of the interaction energy for constant surface charge density boundary conditions on the plates can also be done, which leads to an expression for the interaction potential energy originally given by Usui (1973).

5.6 ELECTROSTATIC INTERACTIONS BETWEEN CURVED GEOMETRIES The electrostatic force or potential energy between two charged bodies, other than parallel plates, is difficult to obtain analytically. In fact, if we note closely, the interaction force per unit area between two parallel plates obtained in the previous section was based on various assumptions about the behavior of electric double layers. Most

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ELECTRIC DOUBLE LAYER

prominently, the closed-form solutions were obtained via the Debye–Hückel approximation, i.e., using the linearized form of the Poisson–Boltzmann equation. A general analytical solution for the electrostatic interaction force between two parallel plates based on the non-linear Poisson–Boltzmann equation, even for the case of symmetric (z : z) electrolytes, is non-trivial, see McCormack et al. (1995). In this context, noting that analytic solutions for the electric potential fields around individual spherical and cylindrical objects were difficult to obtain except for some limiting conditions, it is clearly discernable that rigorous calculations of the electrostatic interaction forces between particles of spherical and other curved geometries based on analytic approaches are still elusive. It is, therefore, becoming more commonplace to utilize numerical procedures for solving the Poisson–Boltzmann equation to obtain the interaction forces for curved geometries. Notwithstanding this fairly recent trend, there have been numerous efforts over the past several decades to explore analytic means for estimating the electrostatic interaction force and potential energy between particles of various geometries. The most common geometries treated were those of two spherical particles, or a spherical particle interacting with a planar surface (infinite flat plate). While the sphere-sphere interactions are of interest in the context of coagulation, the sphere-plate interactions are relevant to particle deposition on surfaces. Before proceeding further with descriptions of some of the most prominent approximate analyses for such systems, let us list the approximations made in most of these approaches, so that it becomes relatively easy to establish the regimes of validity of these solutions. The most common assumptions made in obtaining the electrostatic interaction force and potential energy for particles include: 1. Low surface potentials, and symmetric (z : z) electrolytes, which allow the use of the linearized form of the Poisson–Boltzmann equation, Eq. (5.91). 2. The particle surfaces are maintained either at constant surface potential or constant charge density as they approach each other. 3. Only two particles are interacting in an infinite domain. In other words, the interaction occurs in the limiting case of an infinitely dilute suspension of particles. 4. When two charged particles are at a sufficiently large separation, such that their individual double layers do not substantially overlap, the electric potential distribution around the particles may be obtained by summing the potentials due to the two individual particles. This assumption is generally known as the linear superposition approximation. 5. The surfaces are perfectly smooth. The first three assumptions pertain to the governing Poisson–Boltzmann equation and the imposed boundary conditions. These limiting scenarios were explored in the context of the interaction between two overlapping planar double layers. It is not too difficult to visualize assumption 3 for planar interacting double layers, as the two charged surfaces can be considered as particles with infinitely large radii.

5.6 ELECTROSTATIC INTERACTIONS BETWEEN CURVED GEOMETRIES

157

Assumption 4 refers to the separation distance between the two particles relative to their sizes. The last assumption pertains to the geometry of the particles. It is important to distinguish between these three classes of assumptions, namely, assumptions pertaining to the Poisson–Boltzmann equation, assumptions related to infinitely dilute systems, and assumptions related to the geometry of the interacting surfaces. In fact, ignoring these three classes of assumptions has often led to numerous erroneous analyses of colloidal behavior in the literature. Most analytical expressions for the electrostatic double layer interaction force or potential energy for curved geometries are based on several of the aforementioned assumptions. It should be borne in mind that each of these assumptions, which is valid for certain limiting conditions, can lead to considerably erroneous results when such limiting conditions are violated. For instance, we have seen earlier that the application of the Debye–Hückel assumption at high surface potentials will provide a wrong prediction of the electrostatic potential distribution or the interaction force for planar surfaces. Similarly, if the linear superposition approximation is applied when the particles are sufficiently close such that κr ≫ 1 is violated, the resulting predictions of the potential distribution and the electrostatic force will be erroneous. Such caution should be exercised for every assumption listed above. Therefore, when considering a particular analytic expression for the electrostatic double layer interaction force between two particles, it is extremely important to consider the underlying assumptions that led to such a solution. The use of such a solution should be avoided if any of these assumptions are violated. With this note of caution, we can now proceed with discussing a few approximate expressions for the interaction force between curved geometries. We will primarily limit our attention to two classical techniques that are most commonly used for evaluating the interaction force and potential energy for curved surfaces. These are the Derjaguin approximation (DA) and the linear superposition approximation (LSA). Following this, we will briefly discuss some of the more recent techniques for evaluating the interaction force between colloidal particles of various geometries, including rough surfaces. 5.6.1 The Derjaguin Approximation Derjaguin (1934) recognized that in many cases two macrobodies have significant interaction between them only when the distance of closest approach between these bodies is small compared to the radii of curvature of the macrobodies. Derjaguin proposed for the case of interacting spheres, for small separations compared to the radii of the spheres, that elements on each sphere interact as parallel plane elements at the same separation, Russel et al. (1989). The total sphere-sphere interaction is a sum over the interaction between the planar infinitesimal elements at different distances of separation, Ross and Morrison (1988). We will now expand on the Derjaguin approximation to arrive at a force expression between two interacting spheres. Figure 5.26 illustrates the geometrical approximation needed for developing an expression of force between two spheres separated by distance h at the closest points. Let Fsp be the total force on the spheres and

158

ELECTRIC DOUBLE LAYER

Figure 5.26. Derjaguin’s geometric approximation.

Fp∗ = Fp /Ap be the force per unit area between two infinite parallel plates. Then, the total force between the spheres can be written as  a Fsp = 2π sFp∗ ds (5.149) 0

where s is the radius of a ring-shaped area on the spherical surface, see Figure 5.26. Now, from geometrical considerations, one can write for the separation distance H between the two ring-shaped areas, one on each sphere  H −h = a − a2 − s 2 2

(5.150)

Differentiation of Eq. (5.150) leads to

dH = √

2sds a2 − s 2

(5.151)

In the region of interest, i.e., facing element areas, a 2 ≫ s 2 and Eq. (5.151) reduces to dH =

2s ds a

(5.152)

The force between the spheres then becomes7 Fsp =



h+2a

2π sFp∗

h

or Fsp = π a



a  dH 2s

(5.153)

h+2a

Fp∗ dH

(5.154)

h

It should be noted that in most texts, the upper limit for the integration is taken as infinity. 7

The upper integration limit is traditionally taken as infinity.

5.6 ELECTROSTATIC INTERACTIONS BETWEEN CURVED GEOMETRIES

159

A convenient expression for the electrostatic force per unit area between two interacting plates is given by Eq. (5.111-a) where Fp∗ =

ǫκ 2 ψs2 cosh(κh) + 1

(5.155)

Making use of Eq. (5.155), the electrostatic force between two interacting spheres is then given by  h+2a dH 2 2 Fsp = π aǫκ ψs (5.156) cosh(κH ) + 1 h

Upon integration, one obtains

Fsp =

π aǫκψs2



κH tanh 2

h+2a

(5.157)

h

Recognizing that a ≫ h, the upper limit of the integration can be set to 2a, and setting tanh κa to 1 for κa ≫ 1, one obtains 
κh 2 (5.158) Fsp = π aǫκψs 1 − tanh 2 or Fsp = For κh ≫ 1, one obtains

2π aǫκψs2




exp(−κh) 1 + exp(−κh)

Fsp = 2π aǫκψs2 exp(−κh)



(5.159)

(5.160)

The corresponding interaction potential energy for the force given in Eqs. (5.158) or (5.159) is given as   φsp = 2π aǫψs2 ln 1 + exp(−κh) (5.161) For κh ≫ 1, one obtains

φsp = 2π aǫψs2 exp(−κh)

(5.162)

The general expression for the interaction potential energy of two unequal spheres of radii a1 and a2 having surface potentials ψ1 and ψ2 was given by Hogg et al. (1966). They obtained this expression by applying Derjaguin’s technique to the interaction energy per unit area given by Eq. (5.148). The resulting expression, commonly known as the HHF (Hogg–Healy–Fuerstenau) expression, is given by     π ǫa1 a2 1 + exp(−κh) φsp = 2ψ1 ψ2 ln + (ψ12 + ψ22 ) ln[1 − exp(−2κh)] (a1 + a2 ) 1 − exp(−κh) (5.163)

160

ELECTRIC DOUBLE LAYER

If the two spheres are held at the same surface potential, i.e., ψ1 = ψ2 = ψs , then the above expression simplifies to φsp =

4π ǫψs2 a1 a2 ln[1 + exp(−κh)] (a1 + a2 )

(5.164)

Letting a1 = a2 = a, one reverts to the expression (5.161) derived earlier using the Derjaguin approximation. Although the HHF equation is widely used, its applicability should be considered very carefully. The electric potential must be small enough, i.e., zeψ/(kB T ) < 1, and the radii of the spheres must be large enough as compared to the thickness of the double layer, κa > 10. For Derjaguin’s approximation to be valid, all the interaction energy should come from a small area around the point of closest approach. White (1983) generalized Derjaguin approximation to evaluate the interaction energy between two curved bodies. Let φp∗ (H ) be the energy per unit area associated with half-space (plate) 1 interacting with half-space (plate) 2. In this approximation, the total interaction energy of bodies 1 and 2, φ(h), corresponding to a distance of closest approach h is given by the Derjaguin approximation as  φ(h) = φp∗ (H )dS1 (5.165) S1

where S1 is the surface area of body 1. White very elegantly showed that the interaction energy is given by 2π φ(h) = √ λ1 λ2



∞

(5.166)

φp (H )dH

h

where λ 1 λ2 =




1 1 + ′ R1 R1




1 1 + ′ R2 R2



+ sin2 ϕ




1 1 − R1 R2




1 1 − ′ R1′ R2



(5.167)

Here, R1 and R2 are the principal radii of curvature of body 1, R1′ and R2′ are the principal radii of curvature of body 2, and ϕ is the angle between the principal axes of bodies 1 and 2. By definition, it follows that the force F (h) exerted by one body on another is given by 2π dφ φ ∗ (h) (5.168) =√ F (h) = − dh λ1 λ2 p Equation (5.168) provides a very convenient expression for the evaluation of the interaction force between two bodies from the system geometric properties and the plate-plate interaction energy per unit area. Equation (5.166) and consequently, Eq. (5.168) are valid approximations for any type of interaction energy and they are not restricted to electrostatic interaction

5.6 ELECTROSTATIC INTERACTIONS BETWEEN CURVED GEOMETRIES

161

between surfaces. One can apply Eq. (5.166) with confidence to any given interaction between two bodies provided that the conditions

and

L0 ≪1 R0

(5.169-a)

h ≪1 R0

(5.169-b)

are satisfied, where L0 is the length scale on which the interaction decays to zero and R0 is the smallest radius of curvature of the two bodies. For two interacting spheres of radii a1 and a2 , one obtains R1 = R2 = a1 and R1′ = R2′ = a2 . Using these in Eq. (5.167) gives8
 a1 + a2 2 λ1 λ2 = (5.170) a1 a2 The interaction energy is given by Eq. (5.166) as  ∞ 2π a1 a2 φ(h) = φp∗ (H )dH (a1 + a2 ) h

(5.171)

and the interaction force as F (h) =

2π a1 a2 ∗ φ (h) (a1 + a2 ) p

(5.172)

The interaction energy per unit area between two plates carrying the same electric charge is given by Eq. (5.138-b) as   exp(−κh) φp∗ = 2ǫκψs2 (5.173) 1 + exp(−κh) leading, for the case of two interacting spheres, to  4π ǫκa1 a2 ψs2 ∞ exp(−κH )dH φ(h) = (a1 + a2 ) 1 + exp(−κH ) h Upon integration one obtains φ(h) =

 4π ǫa1 a2 ψs2   ln 1 + exp(−κh) (a1 + a2 )

(5.174)

The expression for the force between two spheres is given by combining Eqs. (5.172) and (5.173) 4π ǫκa1 a2 ψs2 exp(−κh) F (h) = (5.175) (a1 + a2 )[1 + exp(−κh)] 8

In this case, due to symmetry, the angle ϕ = 0.

162

ELECTRIC DOUBLE LAYER

For the case of two equal spheres, a1 = a2 , Eqs. (5.174) and (5.175) revert respectively to Eqs. (5.161) and (5.159). For the configuration of two crossed cylinders of radii a1 and a2 , which is of particular interest due to its use in force measurements, we have R1 = a1 , R1′

= a2 ,

R2 = ∞ R2′

=∞

(5.176-a) (5.176-b)

and ϕ = π/2. Using these parameters, we obtain λ1 λ2 =

1 a1 a2

(5.177)

and the expressions for the interaction between two crossed cylinders are given as √ φ(h) = 2π a1 a2 and



∞

φp∗ (H )dH

(5.178)

h

√ F (h) = 2π a1 a2 φp∗ (h)

(5.179)

White’s approximation approach is very convenient to use to derive expressions of interaction energy and force for curved surfaces. 5.6.2

Linear Superposition Approximation

The basic principle underlying the linear superposition approximation (LSA) is that the electric potential ψ at a location far away from two charged particles can be approximated fairly well as the sum of the potentials due to the individual particles. Consider the geometry shown in Figure 5.27, which depicts two spherical particles with surface potentials ψ1 and ψ2 separated by a distance r. Consider a point P located on the plane S between the two spheres such that its position vectors from the centers of spheres 1 and 2 are given by r1 and r2 , respectively. Provided the point P is sufficiently far away from the centers of either sphere, the linear superposition approximation allows us to write ψP = ψP(1) + ψP(2)

(5.180)

where ψP is the total electrical potential at point P , and ψP(1) and ψP(2) are the electrical potentials at the point due to each of the spheres. In a similar manner, the electric field at location P can be expressed as EP = EP(1) + EP(2) where EP is the total electric field strength at location P .

(5.181)

5.6 ELECTROSTATIC INTERACTIONS BETWEEN CURVED GEOMETRIES

163

Equations (5.180) and (5.181) allow the construction of solutions for the electric potential distribution and field strength for two interacting particles from the corresponding expressions for isolated particles. Once these quantities are determined, it is straightforward to obtain the interaction force by integrating the stress tensor over the intervening plane S in Figure 5.27. Bell et al. (1970) first proposed the LSA technique to obtain the interaction force and potential energy between two spherical particles. They used the solutions for the linearized Poisson–Boltzmann equation around isolated spheres, see Eqs. (5.73-a) and (5.73-b), and obtained the following expression for the magnitude of the interaction force between two spheres of radii a1 and a2
 kB T 2 (1 + κr) F (r) = 4π ǫ exp[−κ(r − a1 − a2 )] (5.182) 1  2 κ 2 a 1 a 2 ze (κr)2 where 1 and 2 are the dimensionless surface potentials of the two spheres, and r is the separation distance between the centers of the two spheres. The force is directed along the line joining the centers of the two spheres. Integrating the force over r yields the interaction potential energy
 kB T 2 a1 a2 φ(r) = 4π ǫ exp[−κ(r − a1 − a2 )] (5.183) 1  2 κ 2 ze r

Figure 5.27. A schematic representation of the linear superposition procedure.

164

ELECTRIC DOUBLE LAYER

A more general form of the LSA expression for the interaction potential is
 a1 a2 kB T 2 exp[−κ(r − a1 − a2 )] (5.184) Y1 Y 2 κ 2 φ(r) = 4π ǫ ze r where Y1 and Y2 are the “effective” surface potentials on the spheres. For small potentials (Debye–Hückel limit), Y1 ≃ 1 and Y2 ≃ 2 . However, for large surface potentials, Y1 and Y2 are more complex functions of the surface potentials. No exact analytical functional forms for these parameters are available, although several approximate relationships have been provided. A common approximation for the effective potential Yi corresponding to large κa is Yi = 4 tanh(i /4)

(5.185)

where i represents the dimensionless potential on sphere i. It should be noted that the linear superposition approximation can be performed for particles of different geometry, provided that analytic expressions for the potential distribution around the isolated particles are available. Furthermore, the solution around an isolated particle need not be based on the linearized Poisson–Boltzmann equation as shown above, but can involve the solution of the nonlinear version of the equation as well. One should, however, exercise caution regarding the use of the LSA expression for the interaction potential, Eq. (5.183). First, the linear superposition approximation restricts its use to the calculation of the interaction potential at large inter-particle separations. Secondly, use of the linearized version of the Poisson–Boltzmann equation restricts the LSA expression to the case of low surface potentials. In fact, it is clearly evident that Eq. (5.183) will provide incorrect results at small separation distances if one of the surface potentials is assigned a value of zero. In this case, the interaction potential predicted by Eq. (5.183) will be zero. However, we noted earlier that the interaction potential between two double layers remains finite at small separations, even when one of the surfaces has zero electrical potential. 5.6.3

Other Approximate Solutions

The Derjaguin and linear superposition approximation methods described in the previous subsections constitute two of the most commonly used leading order approximate techniques that provide simple analytic results for the interaction force and potential energy for particles with curved geometries. Numerous other results have been provided over the years that attempt to improve on the assumptions made in obtaining the interaction force and potential. A common approach was to use higher order series terms resulting from the Taylor expansion of the sinh term in the non-linear Poisson–Boltzmann equation: sinh y ≃ y +

y5 y3 + + ··· 3! 5!

Ohshima et al. (1982) used such a technique to obtain more accurate expressions for the interaction energy between two spheres. White (1977) presented a series solution

5.7 MODELS OF SURFACE POTENTIALS

165

of the Poisson–Boltzmann equation based on a perturbation method. Glendinning and Russel (1983) gave a series solution for the linearized Poisson–Boltzmann equation for two equal sized spheres based on a multipole expansion technique. There has been a growing interest in obtaining estimates of the interaction energy and force between particles of arbitrary shapes, particles and surfaces with roughness, and particles with a distribution of charges. A technique for calculation of interaction energy for particles of any geometry including rough surfaces was recently developed using an extension of Derjaguin’s approach (Bhattacharjee et al., 1998). This technique, called surface element integration (SEI), provides a fairly good approximation for the interaction energy between a particle and a planar surface over a wide range of particle sizes and ranges of interaction. The technique, although numerical, provides a facile route for estimating the interaction energy for surfaces that have arbitrary geometric shapes. The different analytic expressions for the interaction force and energy commonly used in literature have been extensively tabulated in texts (Russel et al., 1989; Elimelech et al., 1995). It should be noted that all analytic solutions will have some form of approximation embedded in it, and hence, one needs to be careful in selecting an expression that is appropriate for a given situation. As mentioned earlier, numerical approaches, based on finite difference or finite element solution of the Poisson– Boltzmann equation, are becoming more common (Carnie et al., 1994; Das et al., 2003) for evaluation of the electric double layer interaction forces. It seems that with enhanced computational infrastructure, such numerical solutions will be used more and more extensively over the coming years. In Chapter 14, we will discuss numerical solution of the Poisson–Boltzmann equation employing finite element analysis. As things stand, one faces considerable challenges in numerically solving the Poisson–Boltzmann equation to a high degree of accuracy. It should be noted that finite difference or finite element techniques are based on approximate formulations of the spatial derivatives of the potential. In finite difference techniques, the derivatives are often based on Taylor expansions, while in finite element methods such formulations are based on piecewise continuous functions (orthogonal polynomials) representing the potential distribution and its spatial derivatives. In light of this, it is not too difficult to realize that the calculation of the stress tensor and the interaction force, which requires the spatial derivative of the electric field (or the second-order derivative of the potential distribution), can be quite erroneous if the numerical discretization is not of high order. The problem is generally tractable for simple geometries like spheres or cylinders. However, for arbitrary particle shapes, numerical solutions of the nonlinear Poisson–Boltzmann equation in its three-dimensional form for asymmetric electrolytes is still considered to be a fairly involved task.

5.7 MODELS OF SURFACE POTENTIALS The surface charge and the surface potential are very much dependent on the surface chemistry (and hence on the origin of the surface charges) and on the ionic concentration in the bulk solution. A number of models have been proposed for

166

ELECTRIC DOUBLE LAYER

predicting the surface potentials in terms of appropriate variables (e.g., Haydon, 1964; Sparnaay, 1972; Chan et al., 1975, 1976; Hunter, 1981, 1991; Takamura and Wallace, 1988). Two such models will be discussed here in order to illustrate the connection between ionic bulk concentration and the surface potential. These models clearly illustrate that by changing the ionic strength, one not only changes the Debye length, but may also change the surface potential. 5.7.1

Indifferent Electrolytes

Let us consider, as an example, silver iodide particles (AgI) suspended in an aqueous solution containing AgI as the major electrolyte. Our goal here is to relate the bulk ionic concentration to the surface potential of the AgI particles by considering the chemical equilibrium between the particle and the solution. First, the difference between the electrochemical potentials µke of the ions of the th k type in the solid phase and in solution is given by µke = µkc + ezk ψ

(5.186)

where zk is the valence of the k th ionic species, ψ is the electrostatic potential difference between the surface and the bulk solution, µkc is the chemical potential difference for the k th species between the surface and the bulk solution, and e is the elementary charge (1.602 × 10−19 C). At equilibrium µke = 0, so that ψ = −

µkc ezk

(5.187)

The above equation relates the difference in the electrostatic potentials between the surface (of the particle) and the bulk solution to the corresponding chemical potential difference between the two. If the electrostatic potential in the bulk solution is taken as zero, then ψ is simply ψs , the surface potential of the solid particle. We shall now consider reasonable “models” for the chemical potentials of the liquid phase and the solid phase so that µkc , and hence, ψs can be estimated. • Let us now assume, for convenience, that the chemical potential of species k in the bulk solution, denoted by µkc,b , can be approximated by that of an ideal solution, i.e., kB T dµkc,b = − (5.188) dnk = −kB T d (ln nk ) nk where nk is the ionic number concentration of the k th ionic species (say, in m−3 ). • For the solid phase, assume that the changes in the concentration of the solution have very little effect on the chemical potentials, µkc,p , of Ag+ and I− (where the subscript p stands for “particle”). Therefore, one can write

 dµkc = d µkc,p − µkc,b ∼ = dµkc,b

(5.189)

5.7 MODELS OF SURFACE POTENTIALS

167

so that d(ψ) = dψs =

kB T d(ln nk ) ezk

(5.190)

Equation (5.190) indicates that a ten-fold increase in nk leads to a relatively small change in the surface potential, a direct consequence of the assumption that the concentrations of the ions in the bulk have a negligible influence on the ionic composition of the particle surface. The ions responsible for the change in the surface potential are called the potential determining ions (pdi). For the system chosen, namely, the AgI solution, the ions are Ag+ and I− . According to Eq. (5.190), the Nernst equation, the interphase potential difference (between the solid and the bulk of the liquid) changes by about 0.06 Volts when the concentration of potential determining ions changes by a factor of 10 (Russel et al., 1989). 5.7.2

Ionizable Surfaces

We shall use a second example with a very different assumption to illustrate the effect of the ionic concentration in the bulk solution on surface potential (Russel et al., 1989). Suppose we have a surface that contains certain molecules that ionize as follows: AH ⇋ A− + H+

(5.191)

The dissociation constant of the above ionization reaction is given by  −  + A S H 0 K= (5.192) [AH]S   where A− S and [AH]S denote the number concentrations (per unit area) at the surface   of the solid phase and H+ 0 denotes the hydronium ion number concentration in the electrolyte solution in the immediate   vicinity of the surface. Note that the surface charge density qs is simply −e A− S . Our task here is to write [A− ]S , and therefore qs , in terms of the surface potential ψs and other measurable quantities such as K and the bulk ionic concentration. Let the total number concentration of the ionizable groups on the surface be   (5.193) [n]S = A− S + [AH]S By combining Eqs. (5.192) and (5.193), one can write  − A S=

K [n]S   K + H+ 0

(5.194)

  The concentration H+ 0 can be related to the concentration of H+ in the bulk, i.e.,  + H b , by recognizing that the ionic concentration follows the Boltzmann distribution,

168

ELECTRIC DOUBLE LAYER

which for a (1 : 1) electrolyte, is 


   eψ H+ = H+ b exp − kB T

(5.195)

Since we have ψ = 0 in the bulk, Eq. (5.195) gives
    + eψs H 0 = H+ b exp − kB T

(5.196)

The above equation relates the ionic concentration in the solution near the surface to that in the bulk solution. Recognizing that the surface charge density is given by qs = −e[A− ]s

(5.197)

combining Eqs. (5.194), (5.196), and (5.197) leads to qs =

−eK[n]s  s K + [H + ]b exp − keψ BT

(5.198)

We still need to eliminate qs from the above equation, i.e., replace qs in terms of ψs , so that ψs can be determined from [H + ]b . We will do this by taking advantage of Eq. (5.52), where
 eψs + 1/2 qs = 2(2ǫkB T [H ]b ) sinh − (5.199) 2kB T Eliminating qs from Eqs. (5.198) and (5.199) yields   
−eK[n] 1 eψs s    = sinh − 2kB T 2(2ǫkB T [H + ]b )1/2 K + [H + ] exp − eψs b

(5.200)

kB T

The surface potential, ψs , is related to the hydronium ion number concentration in the bulk solution, [H + ]b /1000NA . For given [n]s and K values, the solution of Eq. (5.200) provides the surface potential for different values of [H + ]b . Table 5.2 gives the surface potential, ψs , for [n]s of 0.2 × 1018 ions/m2 , which is the equivalent of one ion per 5 nm2 . Different values for the dissociation constant, K, are used in the table. For a given K value, the absolute value of the surface potential decreases with increasing hydronium ion molarity. When the dissociation constant is very large, e.g., K = 102 , [AH ]s becomes negligible and the total number concentration of ionizable groups, [n]s , becomes equal to the surface acidic groups denoted by [A− ]s . In this limiting case, the surface potential is simply given by Eq. (5.52) or Eq. (5.199) where qs = −e[n]s . The values of the surface potential for K = 102 become those denoted by the vertical dashed line of Figure 5.9.

5.8 ZETA POTENTIAL

169

TABLE 5.2. Surface Potential for Different Dissociation Constants and Hydronium Ion Molarities for [n]s = 0.2 × 1018 ions/m2 = 0.2 ions/nm2 . [H + ]b , M

Surface Potential, ψs , mV K = 10

10−4 10−3 10−2 10−1

−4

−68.4 −19.3 −1.3 −0.04

M

K = 10−2 M

K = 102 M

−141.5 −83.0 −30.6 −3.5

−204.9 −145.9 −88.2 −39.7

Note: The units of K in Eq. (5.198) are in ions/m3 . To convert K in units of molarity to a number concentrations multiply by 1000NA .

Models based on more than one dissociation reaction are given by Hunter (1981). Depending on the origin of the charge, other models can be formulated to account for the variation in the surface charge and the ionic strength of the bulk solution. For example, such models can be used to estimate bitumen and solid surface potentials at various levels of pH (Takamura and Wallace, 1988). 5.8 ZETA POTENTIAL In our previous discussion of the Gouy–Chapman treatment of the diffuse electric double layer we have assumed that the ions are point charges and that the inner boundary of the electric double layer is located at the surface of the particle or material under consideration. The location of the outer boundary of the electric double layer is characterized by the inverse Debye length. To this end, the surface potential of a particle in a dielectric medium becomes that “seen” by the fluid surrounding the particle. For a stationary surface, the no-slip velocity occurs at the charged particle surface itself. The Gouy–Chapman treatment of the diffuse double layer runs into some difficulties at small κx values when the surface potential, ψs , is large. Here x is the distance from a charged surface. The coion concentration at the surface can be evaluated from Boltzmann distribution where 
zeψ (5.201) n− = n∞ exp kB T Setting n = 1000NA M, it follows from Eq. (5.201) that
 zeψ Ms = M∞ exp kB T

(5.202)

where M∞ and Ms are the solution molarities far from the surface and at the surface, respectively. For example, at 25◦ C and for a 1 : 1 electrolyte at 0.001 M with a surface potential of ψs = 300 mV, Eq. (5.202) gives a surface molarity for the coions of 118 mol/L,

170

ELECTRIC DOUBLE LAYER

which is not realistic. The inaccuracy of this solution stems from the assumption of point charges and the consequent neglect of the finite ionic diameters (Adamson and Gast, 1997). In real systems, ions are of finite size and they can approach a surface to a distance not less than their radii. Stern (1924) proposed a model in which the electric double layer inner boundary is given by approximately one hydrated ion radius. This inner boundary is referred to as the Stern plane. The gap between the Stern plane and the surface is denoted as the Stern layer. The electric potential changes from the surface potential ψs to a Stern plane potential ψd within the Stern layer and it decays to zero far away from the Stern plane as shown in Figure 5.28. The centers of any ions “attached” to the surface are located within the Stern layer and they are considered to be immobile. Ions whose centers are located beyond the Stern plane form the diffuse mobile part of the electric double layer. Consequently, the mobile inner part of the electric double layer is located between one to two radii away from the surface. This boundary is referred to as the shear plane as shown in Figure 5.28(a). It is on this plane where the no-slip fluid flow boundary condition is assumed to apply. The potential at the shear plane is referred to as the electrokinetic potential, more commonly known as the zeta potential (ζ ). The zeta potential is marginally different in magnitude from the Stern potential ψd as shown in Figure 5.28(b). Shaw (1980) stated that it is customary to identify ψd with ζ and experimental evidence suggests that the introduced errors are small. Electrophoretic potential measurements give the zeta potential of a surface. Although one at times refers to a “surface potential”, strictly speaking, it is the zeta potential that needs to be specified. The Gouy–Chapman treatment of Section 5.2.1 deals with the diffuse electric layer whose inner boundary is the shear plane.

Figure 5.28. (a) Schematic representation of the electric double layer according to the Stern model. (b) Schematic representation of the electric potential profile showing the Debye length, κ −1 , and the overall extent of the electric double layer. The diffuse double layer starts from the Stern plane.

5.9 SUMMARY OF GOUY–CHAPMAN MODEL

171

Figure 5.29. Potential variation with distance for a charged surface: (a) potential reversal due to adsorption of surface active or polyvalent counterions, (b) adsorption of surface active coions.

It should be noted that adsorbed ions can have marked effects on the zeta potential when compared with the surface potential. Adsorbed polyvalent or surface active counterions can cause reversal of charge to occur within the Stern layer as is shown in Figure 5.29(a), where ψs and ζ have different signs. On the other hand, adsorption of surface active coions can create a situation in which ψd has the same sign as ψs and is greater in magnitude than ψs . From the above discussion, it is clear that zeta potential measurements do not give direct information about the surface potential itself when adsorbed ions are present (Shaw, 1980). The interaction of the mobile portion of the diffuse electric layer with an external or induced electric field gives rise to electrokinetic transport phenomena. For example, when an electric field is applied tangentially along a charged surface, the electric field will exert a force on the ions within the mobile diffuse electric layer close to the charged surface resulting in their motion. In turn, the moving ions will drag the surrounding liquid along, thus resulting in the liquid’s flow. The various types of electrokinetic phenomena will be outlined in Chapter 7. 5.9 SUMMARY OF GOUY–CHAPMAN MODEL 5.9.1

Arbitrary Electrolyte

Charge Density: ρf =



zi eni = e

i

 i

zi eψ zi ni∞ exp − kB T

Electroneutrality:  i




zi ni = 0



172

ELECTRIC DOUBLE LAYER

Poisson–Boltzmann Equation:
 e zi eψ ∇ ψ =− zi ni∞ exp − ǫ i kB T 2

Linearized Poisson–Boltzmann Equation, (zi eψ/kB T < 1); Debye–Hückel Solution:   z2 ni∞ eψ e e2 ψ  2 zi ni∞ − i = z ni∞ ∇ 2ψ = − ǫ i kB T ǫkB T i i or ∇ 2ψ = κ 2ψ where κ2 = 5.9.2

2e2  1 2 z ni∞ ǫkB T i 2 i

Symmetrical (z : z) Electrolyte

Charge Density: ρf = ze(n+ − n− ) = −2zen∞ sinh




zeψ kB T



Electroneutrality: n+ − n− = 0 Poisson–Boltzmann Equation: ∇ 2ψ =


 2zen∞ zeψ sinh ǫ kB T

Linearized Poisson–Boltzmann Equation, (zi eψ/kB T < 1); Debye–Hückel Solution:
 2z2 e2 n∞ 2zen∞ zeψ 2 ψ = ∇ ψ= ǫ kB T ǫkB T

or

∇ 2ψ = κ 2ψ where κ2 =

2z2 e2 n∞ ǫkB T

5.10

5.9.3

NOMENCLATURE

173

Forms of Various Notations

Various notations are used in electrostatics literature. The forms of the equations are different depending on whether number or molar concentrations are used. For example, using the molar concentrations (moles/m3 ), one can write the Poisson– Boltzmann equation for a symmetric (z : z) electrolyte as
 zFψ 2zFc∞ 2 sinh ∇ ψ= ǫ RT with 2z2 F 2 c∞ ǫRT where c∞ is the bulk molar concentration of the electrolyte. This type of notation can be easily converted to the notation used in this chapter by recognizing the following relationships: κ2 =

F = eNA R = kB NA

Faraday constant Universal gas constant

n = cNA and Fc = ne

In this respect, we have

zF ze ≡ RT kB T 5.10

e F = kB T RT

NOMENCLATURE

a Ap c c∞ e E E Ex , Ey , Ez F F Fp Fp∗

i.e.,

radius (of spherical particle), m Area of a planar surface, m2 molar concentration, mol/m3 bulk molar electrolyte concentration, mol/m3 elementary charge, C electric field strength, V/m magnitude of electric field, V/m field components along x, y, and z directions in a Cartesian volume, respectively, V/m Faraday number, C/mol force, N interaction force between two planar surfaces, N interaction force per unit area (generally Fp /Ap ) between two planar surfaces, N/m2

174

Fsp g h, H = I I i kB K l M N NA n∞ n+ n− n p p∞ Q Qp qs R r S T = T = Te = Tp u V zi z

ELECTRIC DOUBLE LAYER

interaction force between spheres, N gravitational acceleration, m/s2 separation distance, m unit tensor ionic strength of an electrolyte solution, mol/L unit vector Boltzmann constant (1.38 × 10−23 J/K) dissociation constant, m−3 length, m molar concentration of an electrolyte solution, mol/L number of ionic species Avogadro’s number (6.022 × 1023 mol−1 ) ionic number concentration in the bulk solution, m−3 number concentration of positive ions, m−3 number concentration of negative ions, m−3 unit surface normal vector fluid pressure, Pa fluid pressure in bulk electrolyte, Pa total charge, C total charge on a body, C surface charge density, C/m2 universal gas constant (8.314 × 103 J/(mol.K) radial position, m surface area, m2 absolute temperature, K total stress tensor, N/m2 electric (Maxwell) stress tensor, N/m2 fluid stress tensor (pressure in static fluid), N/m2 fluid velocity vector, m/s volume, m3 valency of ith ionic species absolute value of valency of a (z : z) electrolyte

Greek Symbols ǫ φ(h) κ κ −1 λ1 λ2 µ µk  ψ

dielectric permittivity of medium, C2 /Nm2 or C/Vm electrostatic potential energy at a separation of h between two bodies, J inverse Debye length, m−1 Debye length, m geometrical factor in Derjaguin approximation fluid viscosity, Pa s electrochemical potential difference for species k, J dimensionless electric potential (zeψ/kB T ) electric potential, V

5.11

ψ1 , ψ2 ψd ψs ρ ρf σ ∇ ∇2 ζ 5.11

PROBLEMS

175

surface electric potentials, V electric potential at Stern plane, V true surface electric potential, V fluid material density, kg/m3 volumetric free charge density, C/m3 surface charge density (also denoted by qs ), C/m2 del operator, m−1 Laplacian operator, m−2 zeta potential of a surface, V PROBLEMS

5.1. A 500 ml of 0.02 M KCl is mixed with 300 ml of 0.001 M CuSO4 and 100 ml of 0.001 M AlCl3 at 25◦ C. What is the Debye length for this electrolyte mixture? 5.2. A 500 ml of water was added to 200 ml of 0.1 M KNO3 solution at 25◦ C. What is the Debye length for this solution? 5.3. The exact solution for dimensionless electric potential due to a charged planar surface was provided by Eq. (5.16). Assuming the dimensionless surface potential s ≪ 1, show the exact solution gives Debye–Hückel expression, Eq. (5.24). 5.4. Show that for small surface potentials and for arbitrary electrolytes, the surface charge density of a planar surface is given by qs = ǫκψs where ψs is the surface potential and κ is appropriately defined. 5.5. The solution of the Poisson–Boltzmann equation for the electric potential between two dissimilar planar surfaces held at qa and qb surface charge densities was given by Eq. (5.127) where the constants A2 and B2 are given by Eqs. (5.128) and (5.129), respectively. In order to solve the Poisson–Boltzmann equations, the boundary conditions need to be stated in terms of the surface potentials. (a) State the boundary conditions that are equivalent to constant surface charge densities. (b) Show that for qa = qb = qs , the surface potential is related to the surface charge density by qs ψs = ǫκ tanh(κh/2) 5.6. Evaluate the interaction force between a cylinder of radius a parallel to a plate using White’s approximation. 5.7. In the text, expressions for the interactive electrostatic force between two charged plates having constant surface potentials were provided by Eqs. (5.108)

176

ELECTRIC DOUBLE LAYER

and (5.109) for a z : z electrolyte solution. Making use of the provided force expressions, derive the equivalent expressions for a general electrolyte solution. 5.12

REFERENCES

Adamson, A. W., and Gast, A. P., Physical Chemistry of Surfaces, 6th ed., Wiley-Interscience, New York, (1997). Bell, G. M., Levine, S., and McCartney, L. N., Approximate methods of determining the double-layer free energy of interaction between two charged colloidal spheres, J. Colloid Interface Sci., 33, 335–359, (1970). Bhattacharjee, S., Ko, C.-H., and Elimelech, M., DLVO interaction between rough surfaces, Langmuir, 14, 3365–3375, (1998). Carnie, S. L., Chan, D. Y. C., and Stankovich, J., Computation of forces between spherical colloidal particles – nonlinear Poisson–Boltzmann theory, J. Colloid Interface Sci., 165, 116–128, (1994). Chan, D. Y. C., Perram, J. W., White, L. R., and Healy, T. W., Regulation of surface potential at amphoteric surfaces during particle-particle interaction, J. Chem. Soc. Faraday Trans. 1, 71, 1046–1057, (1975). Chan, D. Y. C., Healy, T. W., and White, L. R., Electrical double-layer interaction under regulation by surface ionization equilibria dissimilar amphoteric surfaces, J.Chem. Soc. Faraday Trans. l, 72, 2844–2865, (1976). Das, P. K., Bhattacharjee, S., and Moussa, W., Electrostatic double layer force between two spherical particles in a straight cylindrical capillary: finite element analysis, Langmuir, 19, 4162–4172, (2003). Derjaguin, B. V., Friction and adhesion. IV: The theory of adhesion of small particles, Kolloid Z., 69, 155–164, (1934). Dube, G. P., Ind. J. Phys., 17, 189, (1943). Dukhin, S. S., and Derjaguin, B. V., Electrokinetic Phenomena, in Surface and Colloid Science, vol. 7, E. Matijevic (Ed.), Wiley, (1974). Elimelech, M., Gregory, J., Zia, X., and Williams, R. A., Particle Deposition and Aggregation: Measurement, Modelling, and Simulation, Butterworth, London, (1995). Everett, D. H., Basic Principles of Colloid Science, Royal Society of Chemistry, London, (1988). Fair, M. C., and Anderson, J. L., Electrophoresis of nonuniformly charged ellipsoidal particles, J. Colloid Interface Sci., 127, 388–400, (1989). Feng, J. J., and Wu, W. Y., Electrophoretic motion of an arbitrary prolate body of revolution toward an infinite conducting wall, J. Fluid Mech., 264, 41–58, (1994). Glendinning, A. B., and Russel, W. B., The electrostatic repulsion between charged spheres from exact solutions to the linearized Poisson–Boltzmann equation, J. Colloid Interface Sci., 93, 95–104, (1983). Gu, Y., The electrical double-layer interaction between a spherical particle and a cylinder, J. Colloid Interface Sci., 231, 199–203, (2000). Haydon, K. A., The electrical double-layer and electrokinetic phenomena, pp. 94–158 in Progress in Surface Science, vol. 1, Danelli, J. F., et al. (Eds.), Academic Press, London, (1964).

5.12

REFERENCES

177

Hiemenz, P. C., and Rajagopalan, R., Principles of Colloid and Surface Chemistry, 3rd ed., Marcell Dekker, New York, (1997). Hogg, R. I., Healy, T. W., and Fuerstenau, D. W., Mutual coagulation of colloidal dispersions, Trans. Faraday Soc., 62, 1638–1651, (1966). Hsu, J., and Liu, B., Solution to the linearized Poisson Boltzmann equation for a spheroidal surface under a general surface condition, J. Colloid Interface Sci., 183, 214–222, (1996). Hunter, R. J., Zeta Potential in Colloid Science, Academic Press, London, (1981). Hunter, R. J., Foundations of Colloid Science, Vol. 1, Oxford University Press, Oxford, (1991). Keh, H. J., and Huang, T. Y., Diffusiophoresis and electrophoresis of colloidal spheroids, J. Colloid Interface Sci., 160, 354–371, (1993). Kittel, C., and Kroemer, H., Thermal Physics, 2nd ed., W. H. Freeman, San Francisco, CA, (1980). Loeb, A. L., Overbeek, J. Th. G., and Wiersema, P. H., The Electric Double-Layer Around a Spherical Colloid Particle, MIT Press, Boston, (1961). Lyklema, J., Fundamentals of Colloid and Interface Science, Vol. II, Academic Press, London, (1995). McCormack, D., Carnie, S. L., and Chan, D. Y. C., Calculations of electric double-layer force and interaction free-energy between dissimilar surfaces, J. Colloid Interface Sci., 169, 177– 196, (1995). McQuarrie, D. A., Statistical Mechanics, Harper and Row, New York, (1976). Ohshima, H., Healy, T. W., and White, L. R., Improvement on the Hogg-Healey-Fuerstenau formulas for the interaction of dissimilar double layers, J. Colloid Interface Sci., 89, 484– 493, (1982). Ohshima, H., Surface-charge density surface-potential relationship for a spherical colloidal particle in a solution of general electrolytes, J. Colloid Interface Sci., 171, 525–527, (1995). Ohshima, H., Surface charge density surface potential relationship for a cylindrical particle in an electrolyte solution, J. Colloid Interface Sci., 200, 291–297, (1998). Ohshima, H., Surface charge density/surface potential relationship for a spherical colloidal particle in a salt-free medium, J. Colloid Interface Sci., 247, 18–23, (2002). Overbeek, J. Th. G., The role of energy and entropy in the electrical double layer, Colloids Surf., 51, 61–75, (1990). Probstein, R. F., Physicochemical Hydrodynamics, An Introduction, 2nd ed., Wiley Interscience, New York, (2003). Ross, S., and Morrison, I. D., Colloidal Systems and Interfaces, Wiley, New York, (1988). Russel, W. B., Saville, D. A., and Schowalter, W. R., Colloidal Dispersions, Cambridge University Press, Cambridge, (1989). Shaw, D. J., Introduction to Colloid and Surface Chemistry, 3rd ed., Butterworths, London, (1980). Sparnaay, M. J., The Electrical Double Layer, Pergamon Press, New York, (1972). Stern, O., Zur theorie der elektrolytischen doppelschicht, Z. Elecktrochem., 30, 508–516, (1924). Takamura, K., and Wallace, D., The physical chemistry of the hot water process, J. Can. Pet. Tech., 27, 98–106, (1988).

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Usui, S., Interaction of electrical double layers at constant surface charge, J. Colloid Interface Sci., 44, 107, (1973). van de Ven, T. G. M., Colloidal Hydrodynamics, Academic Press, London, (1989). Wang, Z. W., Li, G. Z., Guan, D. R.,Yi, X. Z., and Lou, A. J., The surface potential of a spherical colloid particle: Functional theoretical approach, J. Colloid Interface Sci., 246, 302–308, (2002). White, L. R., Approximate analytic solution of the Poisson–Boltzmann equation for a spherical colloid particle, J. Chem. Soc. Faraday Trans. II, 73, 577–596, (1977). White, L. R., On the Deryaguin approximation for the interaction of macrobodies, J. Colloid Interface Sci., 95, 286–288, (1983).

CHAPTER 6

FUNDAMENTAL TRANSPORT EQUATIONS

This chapter reviews the basic equations governing the transport phenomena of interest. First, a very brief summary of the equations relevant to single-component systems (i.e., systems containing only the solvent) is presented. Following this, we consider multicomponent systems in which constituents called solutes are considered explicitly along with the solvent. Typically, the solutes considered in this context will be treated as point masses, examples of which include ions of a dissolved salt. The introduction of ions to the system requires that we consider, in addition to the equations of fluid mechanics and energy or mass transport, the conservation of ionic mass, the conservation of charge, and the current density. Moreover, the momentum equation needs to be modified in this case to account for the electrical forces arising from the presence of ions in the solution. Finally, this chapter presents a brief introduction to the hydrodynamic interactions, which are primarily an outcome of considering finite sized solute particles (colloidal entities) in a multicomponent system. In this context, the modifications of the transport equations required to assess the transport behavior of multicomponent colloidal suspensions will be presented. A number of excellent textbooks are available on the material discussed in this chapter. Bird et al. (1960, 2002) are classic references on transport phenomena. Newman (1991) can be consulted for information regarding transport phenomena in electrochemical systems. The basic vector calculus needed for manipulating the equations of interest is discussed in Aris (1989). A less formal and intuitively more accessible discussion of vector calculus is presented by Schey (1997). The volume by Moon and Spencer (1971)

Electrokinetic and Colloid Transport Phenomena, by Jacob H. Masliyah and Subir Bhattacharjee Copyright © 2006 John Wiley & Sons, Inc.

179

180

FUNDAMENTAL TRANSPORT EQUATIONS

is an excellent reference on coordinate systems and differential equations in thirteen orthogonal systems.

6.1 SINGLE-COMPONENT SYSTEM For systems not undergoing nuclear reactions, one of the basic laws is that of mass conservation. It simply states that for a fixed control volume in space, the rate of mass accumulation within the control volume is the difference between the mass entering and leaving the control volume. In a differential form, the mass conservation law is given by ∂ρ = −∇ · (ρu) (6.1) ∂t The left hand side term is due to mass accumulation and the right hand side term is due to the net mass flux. Equation (6.1) is also referred to as the continuity equation for a fluid in motion. Here, ρ is the fluid density (kg/m3 ) and u (m/s) is the fluid velocity vector relative to a stationary observer. This velocity is measured using a pitot tube or a laser Doppler velocimeter. The symbol t represents time (s) and ∇ is the del operator (m−1 ). When the fluid density is independent of time and space, i.e., steady incompressible flow, the mass conservation equation becomes ∇ ·u =0

(6.2)

The fluid flow of a single-component system under laminar flow conditions with constant density and viscosity is governed by the momentum conservation equation and is given by the Navier–Stokes equation (Bird et al., 2002): ρ

∂u + ρu · ∇u = −∇p + µ∇ 2 u + ρg + fb ∂t

(6.3)

Equation (6.3) represents the force balance on a fluid element in space. Here p is the fluid pressure (Pa or N/m2 ), g is the acceleration due to gravity (m/s2 ), µ is the fluid viscosity (Pa s), and ∇ 2 is the Laplacian operator (m−2 ). The term fb denotes any other body force (force per unit volume) acting on the fluid element in addition to gravity. The gravitational force term, ρg, is also a body force, but is generally written explicitly in the Navier–Stokes equation to differentiate it from the other externally imposed body forces. In electrokinetic transport processes, one needs to consider the electrical body force acting on the fluid. Further discussion on the electrical body force will be presented later in this chapter. Each term of Eq. (6.3) represents a force acting on a unit volume of the fluid. The first term on the left-hand side represents the rate of change of momentum (given by the product of the fluid density and the rate of change of the fluid velocity with time) at a given location within the flow field. For steady state conditions, this term drops out. The second term is due to the fluid inertia and is negligible for low-velocity

6.2 MULTICOMPONENT SYSTEMS

181

(low Reynolds number) flows. The first term on the right-hand side represents the contribution of pressure, the second is due to viscous forces (shear stresses), the third term arises from the body force due to gravity, and the last term arises from any other body force. Equation (6.3) can be written for any orthogonal coordinate system (Bird et al., 2002). The energy equation (convection–diffusion) in the absence of a heat source and energy dissipation is given for constant physical properties (Bird et al., 2002) as 
∂T + u · ∇T = kf ∇ 2 T (6.4) ρCp ∂t where Cp is the fluid specific heat (J/kg ◦ C), T is the temperature of the fluid (◦ C or K), and kf is its thermal conductivity (W/m ◦ C). As an example, we may consider steady two-dimensional incompressible flows with negligible inertia in absence of any body force (except gravity), for which the governing equations become, for the x-directional momentum equation,
2  ∂p ∂ ux ∂ 2 ux =µ + + ρgx (6.5) ∂x ∂x 2 ∂y 2 and, for the y-directional momentum equation, 
2 ∂ uy ∂ 2 uy ∂p =µ + + ρgy ∂y ∂x 2 ∂y 2

(6.6)

for the mass conservation equation, ∂uy ∂ux + =0 ∂x ∂y and, for the energy equation
 
2 ∂T ∂T ∂ 2T ∂ T ρCp ux + uy + = kf ∂x ∂y ∂x 2 ∂y 2

(6.7)

(6.8)

Equations (6.5) to (6.8) are a unique set of equations and are solved subject to appropriate boundary conditions as applicable to the physical problem at hand. From the above discussion, it can be observed that for a single-component fluid one is interested in the fluid velocity, pressure, and temperature. These quantities, characterizing the flow of a single-component fluid, are consequences of different forces acting on the fluid. 6.2 MULTICOMPONENT SYSTEMS 6.2.1

Basic Definitions

While dealing with a single-component system, e.g., flow of water in a pipe, there is no ambiguity as to the meaning of the water velocity at a given location and time.

182

FUNDAMENTAL TRANSPORT EQUATIONS

However, for a fluid made up of many species, and in particular a liquid with dissolved ionic species, the concept of velocity and density needs to be clarified (Bird et al., 2002; Newman, 1991). In a multicomponent system, the local average velocity needs to be defined in terms of the individual velocity of each species that makes up the fluid itself. The local mass average velocity of the fluid is given by n n ρi v i i=1 ρi vi u = n = i=1 ρ i=1 ρi

(6.9)

where u is the local mass average velocity vector of the bulk fluid with respect to a stationary observer as measured by a pitot tube (m/s), vi is the velocity vector of the i th species with respect to a stationary observer (m/s), and ρi is the mass density of the i th species (kg/m3 ) defined as ρi =

mass of species i volume of solution

One can think of ρi as being a mass concentration of the i th species. The density ρ of the solution is given by ρ=

n 

ρi

(6.10)

i=1

The term ρu represents the local rate at which mass passes through a unit area normal to the velocity vector u. Similarly, the term ρi vi represents the local rate at which the i th species mass passes through a unit area normal to the velocity vector vi . A local molar average velocity u∗ may be correspondingly defined as n n ci v i i=1 ci vi u∗ = i=1 = n c i=1 ci

(6.11)

where ci is the molar concentration of the i th species (mol/m3 ) defined as ci =

number of moles of species i volume of solution

The total molar concentration, c, is given by n 

ci = c

(6.12)

i=1

The term cu∗ represents the local molar flux passing through a unit area normal to the velocity vector u∗ . Similarly, the term ci vi represents the molar flux of the i th component.

183

6.2 MULTICOMPONENT SYSTEMS

The molar concentration (mol/m3 ) and mass density (kg/m3 ) are related by ci = 1000

ρi Mi

(6.13)

where Mi is the molar mass (kg/kmol) of the i th species. The mass fraction of the i th species is given by ρi (6.14) wi = ρ The molar (mole) fraction is given by ci c

xi (or yi ) =

(6.15)

The mean or average molar mass of a mixture is given by M = 1000

ρ c

It should be recalled that the mass of a mole of a substance is called the molar mass, M, and its units are kg/kmol. It has the same magnitude as the formerly used molecular weight of g/mol (Probstein, 2003). The factor 1000 appears in order to reconcile the fact that c is given by mol/m3 and not mol/L. Finally, a number concentration ni (m−3 ) can also be defined as ni =

number of particles of species i volume of solution

(6.16)

The number concentration is related to the molar concentration as n i = ci N A

(6.17)

where ci is given in mol/m3 , and NA is the Avogadro number. 6.2.2

Mass Conservation

The mass conservation for a fluid containing several species can be written for an individual species by considering its mass density, velocity, and by taking into account chemical reactions. The mass conservation equation for the i th species in a multicomponent fluid is given by ∂ρi = −∇ · (ρi vi ) + Ri ∂t

(6.18)

where Ri is called the source term, which gives the rate of production of species i due to chemical reaction per unit volume, kg of i th species /m3 s. The species mass

184

FUNDAMENTAL TRANSPORT EQUATIONS

production rate is obtained from the consideration of chemical kinetics. As the overall mass conservation must hold, one can write  Ri = 0 (6.19) Summing for all the species, the mass conservation for a multicomponent system becomes     ∂ ρi Ri (6.20) = −∇ · ρi vi + ∂t Making use of Eqs. (6.9), (6.10), and (6.19), Eq. (6.20) becomes ∂ρ = −∇ · (ρu) ∂t

(6.21)

Equation (6.21) indicates that the mass conservation equation of a multicomponent fluid is identical to that of a single-component fluid as long as its bulk density and its mass average velocity are utilized. In terms of a molar balance, the equivalent to Eq. (6.21) is given by   ∂c = −∇ · cu∗ (6.22) ∂t Equations (6.21) or (6.22) apply to the entire multicomponent system. In practice, many multicomponent systems can be treated as “dilute”, where one of the species (termed solvent) exists in large excess over other species (termed solutes). For instance, the aqueous electrolyte solutions that will be discussed in this book will have small amounts of ions obtained from dissociation of the electrolyte dissolved in a large volume of solvent. For an n-component dilute system, one can write the mass and molar average velocities by separating out the contributions of the solutes and the solvent as n−1 ρi vi ρs vs u = i=1 + ≃ vs ρ ρ

and

∗

u =

n−1 i=1

ci v i

c

+

cs vs ≃ vs c

respectively, where the subscript i refers to the i th solute species and the subscript s denotes the solvent. In other words, ρs , cs , and vs are the mass concentration, molar concentration, and velocity of the solvent, respectively. Thus, for dilute systems, the mass and molar average velocities can be considered identical, i.e., u ≃ u∗ and both velocities are approximately equal to the solvent velocity. Under such conditions, the molar balance Eq. (6.22) can be written as ∂c = −∇ · (cu) ∂t where the mass average velocity is used.

(6.23)

6.2 MULTICOMPONENT SYSTEMS

6.2.3

185

Convection–Diffusion–Migration Equation

When a fluid contains an electrolyte, i.e., charged ionic species, we are interested in the movement or mass transfer of the anions and the cations as well as the bulk fluid. To this end, mass transfer in an electrolytic solution requires a description of the movement of mobile ionic species, material balances, current flow, electroneutrality, and fluid mechanics. The mass flux of the i th species in a moving fluid is given by the product of the density of the i th species ρi and its velocity vi as measured relative to a stationary observer. The mass flux, ρi vi , has the units of kg/m2 s and it is a vector quantity. In a multicomponent system, as is evidenced by Eq. (6.9), the species velocity vi may not be the mass average fluid velocity, u. The deviation of the i th species velocity from the mass average fluid velocity can be due to diffusion if there is a concentration gradient. Furthermore, the i th species velocity can deviate from the mass average fluid velocity due to migration under the influence of an external force if this force acts differently on different species. For instance, gravitational forces acting on species of different densities can be different. In context of charged ions in a solvent, differently charged ions (such as positive and negative) will experience different forces under the same applied external electric field, and will have different velocities. From the preceding discussion, it is evident that all species in a multicomponent system will be transported at a common velocity, namely, the mass average fluid velocity, u, in absence of diffusion and other external forces. Under such conditions, the mass flux of the i th species can be given in terms of the mass average velocity as ji ≡ ji,C = ρi u

(6.24)

where ji,C is termed as the convective flux. This flux is due to the purely convective transport of the species being swept along with the flowing fluid. All other types of fluxes, such as diffusion or migration, are defined in a reference frame that is relative to this convective flux. Noting that any other transport process will change the velocity of the i th species to vi from the mass average fluid velocity, we can write ji = ρi vi = ρi u + ρi (vi − u) = ji,C + ji,Other

(6.25)

where ji,Other refers to the diffusive or migration flux evaluated relative to the mass average fluid velocity. Note that this flux is given as ji,Other = ρi vi′ = ρi (vi − u) where vi′ can be termed as the drift velocity of the species relative to the mass average velocity. Let us now superimpose the effects of diffusion on the flux of a species. In a multicomponent system, the diffusion flux arises from the spatial concentration gradients of different species. To simplify the discussion, we will henceforth assume a dilute solution and restrict the analysis to the solute species only. For a very dilute system,

186

FUNDAMENTAL TRANSPORT EQUATIONS

the mass flux (kg/m2 s) of the i th solute species due to diffusion, with respect to the mass average velocity, u, is governed by Fick’s law as ji,D = −Di ∇ρi

(6.26)

Similarly, the molar flux (mol/m2 s) of the i th solute species due to diffusion, with respect to the molar average velocity, u∗ , is given by j∗i,D = −Di ∇ci

(6.27)

The flux given by Fick’s law is purely due to diffusion as a result of the molecular Brownian motion. The coefficient Di is referred to as the diffusion coefficient of the i th solute species. One can substitute the diffusive mass flux, Eq. (6.26), for ji,Other in Eq. (6.25), and obtain the mass flux of the i th solute species as ji = ρi vi = ji,C + ji,D = ρi u − Di ∇ρi

(6.28)

Equation (6.28) shows that the mass flux of the i th solute species with respect to a stationary observer is a combination of the convective flux due to bulk movement and the diffusional flux. The molar solute flux equation for the i th species can be written as j∗i = ci vi = ci u∗ − Di ∇ci

(6.29)

For a very dilute system, the mass and molar average velocities are approximately equal to each other, i.e., u ≈ u∗ , and we can write Eq. (6.29) as j∗i = ci vi = ci u − Di ∇ci

(6.30)

One can define a number flux (1/m2 s), which is based on number concentration ni , and relate it to the molar flux as ∗ j∗∗ i = NA ji = NA ci vi = ni vi

(6.31)

The number flux, j∗∗ i , can be represented in terms of convective and diffusive transport as j∗∗ i = ni vi = ni u − Di ∇ni

(6.32)

where ni is the number concentration of the i th solute species, expressed in units of m−3 . When deriving the flux equations represented by Eqs. (6.28) to (6.32), the system was assumed to be infinitely dilute. In this context, the dissolved solutes have very small concentrations relative to the solvent. As well, the flux of a species was assumed to take place due to the bulk motion and due to diffusion only. However, when the

6.2 MULTICOMPONENT SYSTEMS

187

dissolved species carry a charge, a transport flux can take place when an electric potential gradient is present. Such a mass transport (or mass transfer) is referred to as migration. This mode of transport is peculiar to a system containing charged species. We need to include a mass transport term due to migration in the flux equations for such charged species. Consider the transport of different solute species in a multicomponent system in presence of an external force, which can act differently on different species. Let us denote the force acting on one solute particle (this can be a molecule or ion) of the i th species by FExt . The particle will move under the influence of this external force, and one can write its velocity as vi′ = ωi FExt

(6.33)

where ωi is defined as the mobility of the particle (velocity per unit applied force). In general, the mobility is given in units of mN−1 s−1 . Note that the velocity of the particle is given as vi′ in Eq. (6.33) signifying that this is the velocity observed solely under the influence of the external force, and is determined in a reference framework of the mass average velocity of the multicomponent system [cf., Eq. (6.25)]. In other words, vi′ = vi − u Multiplying the velocity of the i th species particle obtained from Eq. (6.33) by its mass concentration ρi , we obtain the migration flux relative to the mass average fluid velocity as ji,M = ρi vi′ = ρi ωi FExt (6.34) Substituting this migration flux for ji,Other in Eq. (6.25), we obtain the total mass flux of the i th species under the influence of convection and migration as ji = ρi vi = ji,C + ji,M = ρi u + ρi ωi FExt

(6.35)

Assuming that the contributions to the solute transport due to diffusion and migration can be linearly superimposed, one can write the total mass flux for the i th solute species as ji = ρi vi = ji,C + ji,D + ji,M = ρi u − Di ∇ρi + ρi ωi FExt

(6.36)

Equation (6.36) is widely used in assessing transport behavior of solute species in colloid and electrokinetic transport processes. Note that the unit of the flux is dictated by the unit of concentration used. In Eq. (6.36), the unit of flux is kg/m2 s since the unit of concentration, ρi is kg/m3 . One can write the molar flux (mol/m2 s) as j∗i = ci vi = ci u − Di ∇ci + ci ωi FExt

(6.37)

using the molar concentration ci (mol/m3 ) or the number flux (1/m2 s) as j∗∗ i = ni vi = ni u − Di ∇ni + ni ωi FExt using the number concentration ni (1/m3 ).

(6.38)

188

FUNDAMENTAL TRANSPORT EQUATIONS

At this point, we need to interpret the mobility, ωi , which was introduced in Eq. (6.33). Normally, application of a force to a mass (here a solute particle) causes it to accelerate. In this context, assigning a constant velocity to the particle as in Eq. (6.33) seems counterintuitive. However, it should be noted that as soon as a solute particle starts to move under the influence of the external force, it encounters a counteracting drag force due to the other (predominantly solvent) particles surrounding it. Consequently, in writing Eq. (6.33), it is assumed that the solute particle moves at a terminal velocity attained under the combined influence of the external force and the frictional drag force of the surrounding medium. This is a grossly simplified view of the migration (or drift) velocity, and readers interested in further details of the physics are referred to other sources (Einstein, 1956; Dhont, 1996). In summary, using arguments of the above nature, Einstein showed that ωi =

1 Di = fi kB T

(6.39)

where fi is the inverse of the mobility, and is known as the Stokes–Einstein friction factor, which is given as 6π µa for a spherical particle of radius a. In other words, the mobility of a solute species is proportional to its diffusion coefficient. Substituting Eq. (6.39) in Eq. (6.36), one obtains ji = ρi vi = ji,C + ji,D + ji,M = ρi u − Di ∇ρi +

ρ i Di FExt kB T

(6.40)

Let us now consider the migration term for ionic systems (electrolyte solutions) in presence of an electric field. For a particle of the i th species bearing a charge qi in an electric field E = −∇ψ, the electrical force is given by FEl = qi E = −qi ∇ψ

(6.41)

If the particle is an ion of valence zi , then its charge is given by qi = zi e, and consequently, the electrical force on it will be (6.42)

FEl = −zi e∇ψ

Substituting this electrical force given by Eq. (6.42) for FExt in Eq. (6.40), one obtains ji = ρi vi = ρi u − Di ∇ρi −

zi eρi Di ∇ψ kB T

(6.43)

Equation (6.43) gives the mass flux of the i th ionic species. The first term on the right-hand side of Eq. (6.43) is the flux due to bulk convection, the second term on the right side is due to the concentration gradient (i.e., diffusional process), and the last term is due to migration. The counterpart of Eq. (6.43) in terms of molar flux is given by zi eci Di j∗i = ci u − Di ∇ci − ∇ψ (6.44) kB T

6.2 MULTICOMPONENT SYSTEMS

189

It is also possible to define a flux based on number concentration in terms of the number of ions per unit time per unit area. The corresponding expression is zi eni Di ∇ψ kB T

j∗∗ i = ni u − Di ∇ni −

(6.45)

In traditional electrochemistry, one generally uses the Faraday constant, F, instead of the electronic charge, e, and the universal gas constant, R, instead of the Boltzmann constant, kB . Since F = eNA

(6.46)

R = kB NA

(6.47)

F e = kB T RT

(6.48)

and

one obtains

Using Eq. (6.48) in Eqs. (6.43), (6.44), and (6.45), one obtains the alternate (but completely equivalent) forms of the flux equations as ji = ρi u − Di ∇ρi −

zi Fρi Di ∇ψ RT

(6.49)

j∗i = ci u − Di ∇ci −

zi Fci Di ∇ψ RT

(6.50)

j∗∗ i = ni u − Di ∇ni −

zi Fni Di ∇ψ RT

(6.51)

and

It is also customary in electrochemistry to define the mobility as the velocity of a charge carrier (ion or a charged particle) per unit electric field, or vi′ = −µ∗i ∇ψ

(6.52)

where the ionic mobility, µ∗i , has units of m2V−1 s−1 . Considering the migration terms in Eqs. (6.43), (6.44), and (6.45), as well as in Eqs. (6.49), (6.50), and (6.51), one obtains zi eDi zi FDi µ∗i = = (6.53) kB T RT This equation is referred to as the Nernst–Einstein equation. Comparing the definitions of the mobility given by Eqs. (6.39) and (6.53), we note that µ∗i = zi eωi

(6.54)

190

FUNDAMENTAL TRANSPORT EQUATIONS

One can rewrite the ionic flux equations in an alternate form using the above definition of the ionic mobility, µ∗i . For example, the molar flux given by either Eq. (6.44) or (6.50) can be expressed as j∗i = ci u − Di ∇ci − µ∗i ci ∇ψ

(6.55)

The mass and number fluxes can also be written in a similar manner. Clearly, one can write the ionic flux in various forms. All the forms are inter-related and usually either j∗i or j∗∗ i is used. The flux relationships, Eqs. (6.43), (6.44), and (6.45), as well as the alternate forms given by Eqs. (6.49), (6.50), and (6.51) are called the Nernst–Planck equations. As the mass, molar, or number concentration fluxes represent the i th species crossing a control volume relative to a stationary observer, the conservation laws can be written as ∂ρi = −∇ · ji + Ri (6.56) ∂t ∂ci = −∇ · j∗i + Ri∗ ∂t

(6.57)

∂ni ∗∗ = −∇ · j∗∗ i + Ri ∂t

(6.58)

and

Here, the left hand side of Eqs. (6.56) to (6.58) represents the accumulation rate within a control volume element. The first term on the right hand side is the net flow into the control volume element and the last term is the production rate due to chemical reactions. For steady state with no chemical reactions, Eqs. (6.56) to (6.58) become, respectively, ∇ · ji = 0

(6.59)

∇ · j∗i = 0

(6.60)

∇ · j∗∗ i =0

(6.61)

and

Equations (6.59) to (6.61) constitute the continuity or material balance for the i th species under steady state with no chemical reactions. The appropriate substitution of the flux of the i th species due to convection, diffusion, and migration into the continuity equation leads to the i th species transport equation. For example, substituting for the mass flux of the i th species given by Eq. (6.49) into Eq. (6.56) leads to
 zi FDi ρi ∂ρi = −∇ · ρi u − Di ∇ρi − ∇ψ + Ri ∂t RT

(6.62)

6.2 MULTICOMPONENT SYSTEMS

191

For steady state incompressible fluid flow one can use the fact that ∂ρi /∂t = 0 and ∇ · u = 0 to rewrite Eq. (6.62) for constant Di as zi FDi (∇ · ρi ∇ψ) = Ri RT

u · ∇ρi − Di ∇ 2 ρi −

(6.63)

The first term of Eq. (6.63) represents transport due to convection, the second term represents transport due to diffusion, and the third term represents transport due to migration. Similar transport equations in terms of ci and ni can be written by simple substitution for ρi . In terms of ci , the equivalent to Eq. (6.62) is given by
 ∂ci zi FDi ci = −∇ · ci u − Di ∇ci − ∇ψ + Ri∗ ∂t RT

(6.64)

and the equivalent to Eq. (6.63), where steady state incompressible fluid flow is assumed, is given by u · ∇ci − Di ∇ 2 ci −

zi FDi (∇ · ci ∇ψ) = Ri∗ RT

(6.65)

It should be recalled that for both Eqs. (6.63) and (6.65), it was assumed that the mobility and diffusion of the species are constant, i.e., space independent. This is a reasonable assumption for very dilute systems. Throughout the analysis of this chapter, the assumption of constant ionic mobility and diffusivity will be made. Equations (6.62) to (6.65) are often referred to as the convection–diffusion–migration equations. 6.2.4

Current Density

The flow of current is a result of the individual flux of all the ionic species present in the electrolyte solution. The local current density vector (A/m2 ) is given by i=F or i=e





zi j∗i

(6.66)

zi j∗∗ i

(6.67)

In terms of ionic molar concentration, Eqs. (6.50) and (6.66) provide the current i = Fu



zi ci − F



zi ni − e



Di zi ∇ci −



Di zi ∇ni −

 F2 ∇ψ zi2 Di ci RT

(6.68)

e2 ∇ψ  2 zi Di ni kB T

(6.69)

Making use of Eqs. (6.45) and (6.67), one can write the current density in terms of the ionic number concentration i = eu

192

FUNDAMENTAL TRANSPORT EQUATIONS

Equation (6.68) or Eq. (6.69) indicates that the current density is due to three contributions. The first term gives the contribution due to convection, the second term gives the contribution due to diffusion where there is an ionic concentration gradient, and the third term is the contribution due to migration where there is an electric potential gradient. The relative contribution due to a given mechanism is much dependent on the physical situation at hand. For the special case of an electrically neutral electrolyte solution, the first term on the right-hand side of Eq. (6.69) or Eq. (6.68) drops out since   z i ni = zi ci = 0 (6.70) This is equivalent to stating that the bulk motion of a fluid with a zero volume charge density does not contribute to the current density. When there is no concentration gradient and the electrolyte solution is electrically neutral, we consider only the last term on the right side of Eq. (6.68) to obtain i=−

F 2  2 zi Di ci ∇ψ RT

(6.71)

Setting σ to be the electric conductivity of the solution, one can write i = −σ ∇ψ

(6.72)

where F2  2 e2  2 zi Di ci = zi Di ni (6.73) RT kB T in units of S/m, where S has the units of Siemens = Ampere/Volt. For a symmetric (z : z) electrolyte, and assuming the diffusivity of both ions are identical, i.e., Di = D, Eq. (6.73) becomes 2F 2 z2 Dc∞ 2e2 z2 Dn∞ σ = = (6.74) RT kB T where c∞ and n∞ are the bulk molar and number concentration of the ions, respectively. In Chapters 8, 9, and 10, we will denote the bulk electrolyte solution conductivity, σ , given by Eq. (6.73), as σ ∞ . This will be done to distinguish the bulk electrolyte solution conductivity from the conductivity of a concentrated suspension of charged particles. The bulk electrolyte conductivity is different from the suspension conductivity as will be seen in Chapter 9. One can consider the bulk electrolyte conductivity to be the same as the suspension conductivity in the limiting case when the particle volume fraction in the suspension approaches zero (φp → 0). Equation (6.72) is an expression of Ohm’s law valid for an electrically neutral electrolyte in the absence of concentration gradients (Newman, 1991). The electric conductivity of the i th species can be written as σ =

σi =

e2 2 F2 2 z i D i ci = z D i ni RT kB T i

193

6.2 MULTICOMPONENT SYSTEMS

In order to compare electrolyte electric conductivities, it becomes necessary to normalize the ionic conductivity with the i th species concentration. We can define a molar electric conductivity, λi , for an ionic species (Sm2 /mol) as F2 2 σi = z Di ci RT i

λi =

(6.75)

This is the conductivity of an ionic species for one mole of the ions in one cubic meter of the solution. In some literature, a term called the equivalent electric conductivity is used, which is defined as λi /zi . One should be careful in reading tables for the conductivity, since the symbols used can be defined differently. The electric conductivity of a solution is given by σ =



(6.76)

λi ci (S/m)

For a single salt electrolyte, we have (6.77)

σ = λ+ c+ + λ_ c_

One can also define a molar electric conductivity, σM (Sm2 /mol), of an electrolyte in terms of the individual molar electric conductivities of the dissociated ions using σM = ν+ λ+ + ν− λ− , where ν+ and ν− are the respective stoichiometric numbers of cations and anions formed by dissociation of the electrolyte. For example, the molar electric conductivity of Na2 SO4 given in Table 6.2 at 298 K and infinite dilution can be closely matched using the molar electric conductivities of the Na+ and SO2− 4 ions (Table 6.4) as σM = 2λ+ + 1λ− = (2 × 5.01 + 16)10−3 = 2.602 × 10−2 Sm2 /mol Both λi and σM depend on electrolyte concentration, and one generally uses the o symbols λoi and σM to represent the molar ionic and molar electrolyte conductivities at infinite dilution, respectively. It should be emphasized here that σM refers to the TABLE 6.1. Standard Solutions for Calibrating Conductivity Measurement Cells. Grams of KCl per kg H2 O 76.5829 7.47458 0.74582 7.45510 (0.1 M) 0.74551 (0.01 M)

Solution Conductivity, σ , (S/m) 0◦ C

18◦ C

25◦ C

6.514 0.7134 0.07733

9.781 1.1163 0.12201

11.131 1.2852 0.14083

25◦ C* 1.2854 0.14086 1.28217 0.14079

Source: Data are from Marsh (1987) except for the last column (*), which are from Wu et al. (1989).

194

FUNDAMENTAL TRANSPORT EQUATIONS

o TABLE 6.2. Molar Electric Conductivity, σM (σM at infinite dilution), of Electrolytes in Aqueous Solutions.

Compound

AgNO3 1/2BaCl2 1/2CaCl2 1/2Ca(OH)2 1/2CuSO4 HCl KBr KCl KI KNO3 KOH LiCl 1/2MgCl2 NH4 Cl NaCl NaI NaOH 1/2Na2 SO4 CH3 COONa 1/2ZnSO4

Concentration, mol/L Infinite Dilution

0.0005

0.001

0.005

0.01

0.02

0.05

0.1

133.29 139.91 135.77 258 133.6 425.95 151.9 149.79 150.31 144.89 271.50 114.97 129.34 149.60 126.39 126.88 247.70 129.80 91.00 132.70

131.29 135.89 131.86 — 121.6 422.53 — 147.74 148.2 142.70 — 113.09 125.55 147.5 124.44 125.30 245.50 125.68 89.20 121.30

130.45 134.27 130.30 — 115.2 421.15 148.9 146.88 143.32 141.77 234.0 112.34 124.15 146.70 123.68 124.19 244.60 124.09 88.50 114.47

127.14 127.96 124.19 233 94.02 415.59 146.02 143.48 144.30 138.41 230.0 109.35 118.25 134.40 120.59 121.19 240.70 117.09 85.68 95.44

124.70 123.88 120.30 226 83.08 411.80 143.36 141.20 142.11 132.75 228.0 107.27 114.49 141.21 118.45 119.18 237.90 112.38 83.72 84.87

121.35 119.03 115.59 214 72.16 407.04 140.41 138.27 139.38 132.34 — 104.60 109.99 138.25 115.70 116.64 — 106.73 81.20 74.20

115.18 111.42 108.42 — 59.02 398.89 135.61 133.30 134.90 126.25 219.0 100.06 103.03 133.22 111.01 112.73 — 97.70 76.88 61.17

109.09 105.14 102.41 — 50.55 391.13 131.32 128.90 131.50 120.34 213.0 95.81 97.05 128.69 106.69 108.73 — 89.94 72.76 52.61

The values of the molar electric conductivities are given in 10−4 Sm2 mol−1 at a temperature of 25◦ C. The electrolyte solution conductivity is evaluated as the product of the molar conductivity of the solution and the solution molarity. Source: Adapted from D. R. Lide (Ed.), CRC Handbook of Chemistry and Physics, CRC Press, 2002. Comprehensive data on electric conductivity and diffusivity are also provided in CRC Handbook of Chemistry and Physics.

molar electric conductivity of the electrolyte and is not to be confused with the solution electric conductivity, σ . At infinite dilution, i.e., when the electrolyte concentration approaches zero, the molar conductivity σM approaches a maximum while the solution conductivity σ approaches zero. The electric conductivity values are sensitive to the solution temperature, and some standard values are provided in Tables 6.1 to 6.4. In industrial practice, for convenience, the electric conductivity is expressed in µS/cm. Figure 6.1 gives an estimate of electric conductivity of aqueous solutions. For ultrapure water, the electric conductivity is about 0.055 µS/cm and for good quality raw water it is about 50 µS/cm. In some industries, when the conductivity of water approaches that of ultrapure water, electric resistivity is used instead. It is the

195

6.2 MULTICOMPONENT SYSTEMS

TABLE 6.3. Values of Equivalent Electric Conductivities, λi /zi , and Diffusion Coefficients of Selected Ions at Infinite Dilution in Water at 25◦ C (Newman, 1991). Cation H+ Li+ Na+ K+ NH+ 4 Ag+ Tl+ Mg2+ Ca2+ Sr2+ Ba2+ Cu2+ Zn2+ La3+ Co(NH3 )3+ 6

zi 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3

λi /zi × 104 Sm2 /mol

Di × 109 m2 /s

349.8 38.69 50.11 73.52 73.4 61.92 74.7 53.06 59.50 59.46 63.64 54.0 53.0 69.5 102.3

9.312 1.030 1.334 1.957 1.954 1.648 1.989 0.7063 0.7920 0.7914 0.8471 0.72 0.71 0.617 0.908

Anion

zi

λi /zi × 104 Sm2 /mol

Di × 109 m2 /s

OH− Cl− Br− I− NO− 3 HCO− 3 HCO− 2 CH3 CO− 2 SO2− 4 Fe(CN)3− 6 Fe(CN)4− 6 IO− 4 ClO− 4 BrO− 3 HSO− 4

−1 −1 −1 −1 −1 −1 −1 −1 −2 −3 −4 −1 −1 −1 −1

197.6 76.34 78.3 76.8 71.44 41.5 54.6 40.9 80.0 101.0 111.0 54.38 67.32 55.78 50.0

5.260 2.032 2.084 2.044 1.902 1.105 1.454 1.089 1.065 0.896 0.739 1.448 1.792 1.485 1.330

reciprocal of the electric conductivity. For ultrapure water, the electric resistivity is about 18.2 M cm. The effect of the electrolyte concentration on the molar electric conductivity for a symmetric (1 : 1) electrolyte solution can be given by √ o o σM = σM − (A + BσM ) M for M < 10−3 o with A = 0.00602 and B = 0.229. Here σM and σM are the molar electric conductivities at an infinite dilution and at a molarity of M. The molar electric conductivity is given in Sm2 /mol. The relationship between the electric conductivity and molar conductivity of an electrolyte solution is given by

σ = 1000MσM One can think of the difference between the molar electric conductivity at a given electrolyte concentration and the molar electric conductivity at infinite dilution as being the effect of the electrolyte concentration, i.e., finite concentration. In general, the electric conductivity of an electrolyte solution increases with temperature. For moderately high conductive electrolyte solutions, a rough estimate for the change in electric conductivity, σ = σ2 − σ1 , over a temperature change, T = T2 − T1 , is given by K = (σ/T )/σ1 , where T is in degrees Celsius. The

196

FUNDAMENTAL TRANSPORT EQUATIONS

TABLE 6.4. Molar Electric Conductivity, λi , of Some Ions at Infinite Dilution in Water (Atkinson, 1972). Temperature, T K Cation H+

Li+ Na+ Ca2+ Cu2+ La3+ Anion OH− Cl−

Br−

NO− 3 HCO− 3 SO2− 4

◦

C

Molar conductivity λi × 103 Sm2 mol−1

288 298 308 298 288 298 308 298 298 298

15 25 35 25 15 25 35 25 25 25

30.1 35.0 39.7 3.87 3.98 5.01 6.15 11.9 10.8 20.0

298 288 298 308 288 298 308 298 298 298

25 15 25 35 15 25 35 25 25 25

19.8 6.14 7.63 9.22 6.33 7.83 9.42 7.14 4.45 16.0

interpolation formula is given as σ = σ1 [1 + K(T − T1 )] where T > T1 and σ1 is the conductivity at T1 . Values of K for different electrolytes are tabulated in Table 6.5.

Figure 6.1. Electric conductivities of water under different conditions.

6.2 MULTICOMPONENT SYSTEMS

197

TABLE 6.5. Values of the Parameter K for Different Electrolytes. Electrolyte

K (×102 )

Acids Bases Salts Fresh water

(1.0–1.6) (1.8–2.2) (2.2–3.0) 2.0

1/◦ C 1/◦ C 1/◦ C 1/◦ C

EXAMPLE 6.1 Evaluation of Current Density. Evaluate the current density for a 0.01 molar NaCl solution at 298 K for a field strength Ex of 1000 V/m. Assume electrical neutrality and uniform ionic concentration. Solution The current density equation is given by ix = −σ

dψ dx

where x is the current flow direction. From Table 6.2, the molar conductivity is given by 118.45 × 10−4 Sm2 /mol    Sm2 V kmol 1000 mol ix = − 118.45 × 10−4 × −1000 0.01 3 × mol m kmol m ix = 118.45

A m2

EXAMPLE 6.2 Electric Conductivity. Evaluate the solution electric conductivity and molar conductivity of a 0.0005 M solution of Na2 SO4 at 25◦ C using the ionic equivalent conductivities at infinite dilution given in Table 6.3. Solution From Table 6.3, at infinite dilution: For Na+ , λ = 50.11 × 10−4 Sm2 /mol z For SO2− 4 , λ = 80.0 × 10−4 Sm2 /mol z

198

FUNDAMENTAL TRANSPORT EQUATIONS

The electrolyte solution conductivity is given by σ =



λi ci = (50.11 × 10−4 × 1)(1000 × 2 × 0.0005) + (80.0 × 10−4 × 2)(1000 × 0.0005)

or σ = 0.005011 + 0.008 = 0.0130 S/m The molar conductivity of the electrolyte is given by σM =

0.0130 = 0.0260 Sm2 /mol 0.0005 × 1000

From Table 6.2, the molar conductivity at infinite dilution is given as 2 × 129.80 × 10−4 = 0.02596 Sm2 /mol. As would be expected, this value is very close to that obtained using Table 6.3. However, the actual value using Table 6.2 for 0.0005 mol/L is (125.68 × 10−4 × 2)(0.0005 × 1000) = 0.02514 Sm2 /mol This value is slightly lower than that at infinite dilution. In our calculations, we used the individual values for λi /zi at infinite dilution. 6.2.5

Conservation of Charge

It is a physical law of nature that electric charge is conserved. This is implicitly built into the basic transport relationships. Multiplying Eq. (6.57), the equation for the conservation of the i th ionic species, by zi F yields ∂ F zi ci = −F∇ · (zi j∗i ) + Fzi Ri∗ (6.78) ∂t For convenience, zi was included inside the differentiation operator. One can now sum over all species to obtain F

  ∂  zi ci = −F∇ · zi j∗i + F zi Ri∗ ∂t

(6.79)

Recognizing that the first term on the right hand side is related to the current density, [see Eq. (6.66)], one can write Eq. (6.79) as F

 ∂  zi ci = −∇ · i + F zi Ri∗ ∂t

(6.80)

Equation (6.80) represents the conservation of the current density under unsteady state conditions with chemical reactions.

6.2 MULTICOMPONENT SYSTEMS

199

The last term of Eq. (6.80) is zero as long as all homogeneous reactions that give rise to the Ri∗ term are electrically neutral. With the latter in mind, Eq. (6.80) becomes F

∂  zi ci = −∇ · i ∂t

(6.81)

 Noting that the term F zi ci represents the volumetric free charge density (C/m3 ), one can relate this term to the electric field using the Gauss law [cf., Eq. (3.110)]. This gives  F zi ci = ∇ · (ǫE) (6.82) where E = −∇ψ is the electric field. Substituting Eq. (6.82) in (6.81) yields 
∂ (ǫE) = −∇ · i (6.83) ∇· ∂t or


 ∂(ǫE) ∇· +i =0 ∂t

(6.84)

∇ ·i=0

(6.85)

The first term in Eq. (6.84) is called the displacement current. The terminology stems from the fact that for linear dielectric materials, ǫE = D, where D is the electric displacement vector [see Eq. (3.109)]. The second term in Eq. (6.84) is referred to as the transport current. Under steady state conditions, the displacement current vanishes, leading to Equation (6.85) is the mathematical expression of the conservation of charge or simply the continuity of the current density under steady state conditions. 6.2.6

Binary Electrolyte Solution

The convection–diffusion–migration transport equation given by Eq. (6.64) as well as the current density given by Eq. (6.68), are valid for a very dilute system containing ionic species derived from several salts. As an example, solutions containing Na2 SO4 and CaCl2 individually or as a mixture can be represented by these equations. For the special case of electroneutrality with no chemical reaction and for a solution containing a single salt, e.g., Na2 SO4 , the transport and current density equations can be greatly simplified. For clarity, let us denote the positive ions by the symbol “+” and the negative ions by the symbol “−”. Next, we introduce γ+ and γ− as the number of positive ions and negative ions produced from the dissolution of one molecule of the single salt under consideration. The electroneutrality condition, given by z+ c+ + z− c− = 0

(6.86)

c+ c− = γ+ γ−

(6.87)

can be replaced by

200

FUNDAMENTAL TRANSPORT EQUATIONS

We introduce a scaled ionic concentration C using Eq. (6.87) as C=

c− c+ = γ+ γ−

(6.88)

In terms of the scaled ionic concentration, the transport equation, Eq. (6.64), under the assumption of Ri∗ = 0, electroneutrality, and incompressible flow, can be written for each ionic species as ∂C F + u · ∇C = D+ ∇ 2 C + z+ D+ ∇ · (C∇ψ) ∂t RT

(6.89)

and ∂C F + u · ∇C = D− ∇ 2 C + z− D− ∇ · (C∇ψ) (6.90) ∂t RT The divergence term containing ∇ · (C∇ψ) can be eliminated from Eqs. (6.89) and (6.90) leading to ∂C + u · ∇C = Deff ∇ 2 C (6.91) ∂t where (z+ − z− )D+ D− Deff = (6.92) z+ D+ − z − D− Equation (6.91) governs the scaled ionic concentration of the single salt species in a solution where convective, diffusive, and migration fluxes can be present. Here, Deff is the effective diffusion coefficient. Equation (6.91) is known in the mass transfer literature as the convection–diffusion equation whose solution is given for different geometries and boundary conditions.Apart from the assumption of a single salt system and no chemical reaction, the serious restriction is the electroneutrality assumption. We need to keep in mind, however, that Eq. (6.91) was derived without assuming a zero current density. Although the current density is provided through Eqs. (6.68) or (6.85), a simplified expression can be obtained for the single salt case within the restrictions of no chemical reaction and electroneutrality. Introducing the scaled ionic concentration into Eq. (6.68) leads to

2 

 F2 2 C∇ψ z+ D+ γ+ + z− D− γ − i = −F D+ z+ γ+ + D− z− γ− ∇C − RT

(6.93)

Using Eqs. (6.86) and (6.87) to eliminate c+ and c− leads to z+ γ+ = −z− γ−

(6.94)

Making use of the relationship (6.94), Eq. (6.93) becomes F i = −(D+ − D− )∇C − (z+ D+ − z− D− ) C∇ψ z+ γ+ F RT

(6.95)

6.2 MULTICOMPONENT SYSTEMS

201

Equation (6.95) provides the current density for an electroneutral system with no chemical reactions. In the absence of a current density, i.e., i = 0, Eq. (6.95) gives (D+ − D− ) F ∇ψ = − ∇ ln C RT z+ D+ − z− D−

(6.96)

where ∇C = ∇ ln C C Equation (6.96) shows the relationship between the electric field and concentration gradients when there is no current density at a given location within the system for an electroneutral binary electrolyte with no chemical reactions. It should be noted that the flux equations were derived on the assumption that the solution is dilute and that the concentrations are used to establish the driving forces. However, for a semi-dilute solution (with moderate salt concentrations), the gradient of the electrochemical potential is used as the driving force for diffusion and migration. The equivalent of the Nernst–Planck equation (in molar concentration units) becomes j∗i = ci u − ωi ci ∇µi (6.97) where µi is the electrochemical potential of a particle of the i th species, and ωi is the mobility defined as velocity of the i th species per unit driving force. Here, the driving force is obtained from the gradient of the electrochemical potential, comprising of the chemical potential and the electrical potential,
 zi eψ ∇µi = ∇ µi,c + kB T The electrochemical potential µi depends on the composition of the electrolyte solution, represented by the chemical potential term µi,c , and the local electrical potential energy. The local electrical potential energy depends on the charge of a particle of the i th species and the electric potential, ψ. The electrical potential energy is given by the term zi eψ/kB T . For more details, the reader is referred to Newman (1991). 6.2.7

Boltzmann Distribution

The Boltzmann distribution was previously derived in Chapter 5 using heuristic arguments. One can make use of the Nernst–Planck relationship to derive the Boltzmann distribution for the ionic number concentration near a charged surface. Consider a flat surface whose normal is in the x-direction. Equation (6.45) for the ionic flux gives zi eni Di dψ dni − (6.98) jix∗∗ = ni ux − Di dx kB T dx where the subscript x denotes the x-direction. At equilibrium, we have zero fluid velocity, ux = 0, and zero flux of ions, jix∗∗ = 0. Denoting the electric potential under

202

FUNDAMENTAL TRANSPORT EQUATIONS

these stationary (no flow) conditions by the symbol ψ eq , Eq. (6.98) becomes  dni zi eni dψ eq + =0 (6.99) dx kB T dx or zi e dψ eq d(ln ni ) + = 0. dx kB T dx

(6.100)

Let ni = ni∞ eq ψ∞

eq at ψ eq = ψ∞

(6.101)

Here, is the reference potential measured in a stationary (no flow) electroneutral electrolyte solution far away from the charged surface. The solution of Eq. (6.100) subject to boundary condition Eq. (6.101) is given by    eq  zi e ψ eq − ψ∞ ni = ni∞ exp − (6.102) kB T eq

For convenience, one may set the reference potential ψ∞ = 0 to obtain 
zi eψ eq ni = ni∞ exp − kB T

(6.103)

eq

Here we recover the Boltzmann distribution. Note that setting ψ∞ = 0 does not necessarily mean that the electric potential is actually zero far away from the charged surface (in the stationary electroneutral electrolyte solution). One should interpret the potential ψ eq in the Boltzmann distribution equation (6.103) as the difference between the actual electric potential near a charged surface and the electric potential in a stationary electrolyte reservoir located at an infinite distance from the surface. It should be noted that the Nernst–Planck relationship gives the Boltzmann distribution with the explicit assumptions of j∗∗ i = 0 and u = 0. In a flow system, the above assumptions are not strictly valid. The electric double layer theory discussed in Chapter 5 was based on the Boltzmann distribution, which led to the formulation of the Poisson–Boltzmann equation. The Poisson–Boltzmann equation is a special form of the Poisson equation, and is only applicable under stationary (no flow and no ion movement) conditions. Thus, when considering a general electrokinetic transport problem, the Poisson–Boltzmann equation should be treated as the equation governing the stationary (no flow) electric potential distribution, and may be expressed as  e zi eψ eq 2 eq ∇ ψ =− zi ni∞ exp − (6.104) ǫ i kB T where ψ eq denotes the electric potential distribution at equilibrium near a charged surface placed in a stationary electrolyte reservoir. Any non-zero ionic flux (j∗∗ i  = 0),

203

6.2 MULTICOMPONENT SYSTEMS

caused by whatsoever external means, will change the ion concentration distributions from the stationary Boltzmann distribution. In such dynamic situations, the total electric potential (generally represented as ψ) will differ from ψ eq . This total electric potential, ψ, in the dynamic situation can strictly be determined from a coupled solution of the Poisson, Nernst–Planck, and Navier–Stokes equations. In many electrokinetic transport problems, the difference between the total potential under dynamic conditions, ψ, and the equilibrium potential, ψ eq , is small, and can be related through a perturbation variable as ψ = ψ eq + δψ

(6.105)

where δψ is a linear perturbation in the potential caused by ionic movement. We will discuss application of this perturbation approach in Chapter 9 in context of electrophoresis. In summary, when addressing electrokinetic transport processes, one must always keep in mind that the electric potential employed in the dynamic problem is the “total” potential, ψ, and not the equilibrium or stationary potential, ψ eq , obtained from a solution of the Poisson–Boltzmann equation. In this chapter, whenever we use the symbol ψ, it denotes the total electric potential in a dynamic condition. 6.2.8

Momentum Equations

As was discussed earlier, the momentum equation given by Eq. (6.3) contains a generic body force term, fb , which can be used to consider any type of force acting on a fluid volume element. In electrokinetic transport processes, this body force arises due to an electric field. The electric body force per unit volume on the fluid (N/m3 ) is given by 1 1 fE = ρf E − E · E∇ǫ + ∇ 2 2


  ∂ǫ ρ E·E ∂ρ T

(6.106)

where ρ is the fluid mass density and ρf is the free charge density (Russel et al., 1989; Verbrugge and Pintauro, 1989). The force term given by Eq. (6.106) is for a variable dielectric constant. For the special case of a constant permittivity, the electrical body force per unit fluid volume, fE , due to the electric field becomes fE = ρ f E

(6.107)

Recognizing that the electric field, E, is related to the electric potential, ψ, by E = − ∇ψ

(6.108)

fE = −ρf ∇ψ

(6.109)

one can write for the force term

204

FUNDAMENTAL TRANSPORT EQUATIONS

where the units of ρf are C/m3 and the units of ∇ψ are V/m. Introducing the electric force term for the body force, fb , in Eq. (6.3), we obtain ρ

∂u + ρu · ∇u = −∇p + µ∇ 2 u + ρg − ρf ∇ψ ∂t

(6.110)

Equation (6.110) is the momentum equation in the presence of an electric body force. It is the modified Navier–Stokes equation when an electric force is considered. It should be stressed here that ψ in Eq. (6.110) represents the total electric potential. The free charge density is given by   zi ni = F zi ci (6.111) ρf = e where e is the fundamental charge, zi is valence of the i th species, ni is the ionic concentration, F is the Faraday constant, and ci is the molar concentration. For steady-state creeping flow where inertia is neglected, Eq. (6.110) becomes ∇p = µ∇ 2 u + ρg − ρf ∇ψ

(6.112)

The mass conservation is still given by Eq. (6.2). For steady-state and constant density, one obtains ∇ ·u=0

(6.113)

Equations (6.112) and (6.113) are the equations governing the fluid mechanics for the transport of an electrolytic solution under steady-state creeping flow conditions. Additional information is needed to provide closure to these equations. To this end, the electric potential ψ is given by the Poisson equation and the species conservations are provided by the convection–diffusion–migration equation for every ionic species, Eq. (6.64). The Navier–Stokes equation can be solved after setting appropriate boundary conditions corresponding to a given flow situation. The boundary conditions can specify the velocities or pressures at given locations of the flow boundary. Of particular interest with respect to boundary conditions for these equations is the no-slip condition, which is generally applied at the solid boundaries. The no-slip condition implies that the fluid in contact with a solid object moves at the same velocity as the object. If the object is stationary, such as the wall of a container, then the fluid velocity is zero at the wall. Imposition of the no-slip condition usually sets up a velocity gradient in the fluid near the solid-fluid interfaces. Such a gradient gives rise to a viscous stress. The local fluid stress tensor at a point on the surface of the solid object is given by,

 = = τ = −pI + µ ∇u + (∇u)T (6.114) =

where p is the pressure, I is the unit tensor, and the final term on the right hand side represents the viscous stresses. This fluid stress can be integrated over the surface of the solid object to determine the total hydrodynamic force acting on the object in a manner analogous to the integration of the Maxwell stress to provide the electrical force acting on a body.

6.3 HYDRODYNAMICS OF COLLOIDAL SYSTEMS

205

6.3 HYDRODYNAMICS OF COLLOIDAL SYSTEMS So far in this chapter, we focussed on the transport behavior of a solute particle (ion) assuming it to be a point mass. In other words, the solute particle did not have a finite size, and in the purely hydrodynamic limit, when diffusion or migration fluxes are absent, the solute particle velocity was assumed to be equal to the mass average velocity of the fluid, vi = u

(6.115)

When considering colloidal particles, it is generally not prudent to consider them as point masses. For such particles, in the purely hydrodynamic limit (in the absence of diffusion and migration), the particle velocity can differ from the fluid velocity. Denoting the particle hydrodynamic velocity vector as vp to distinguish it from the velocity vector, vi , which was used to define the overall particle velocity arising from the combined influence of convection, diffusion, and migration, we note that several flow situations can give rise to conditions when vp  = u

(6.116)

In this section, we discuss the mechanisms causing the above mentioned deviation of the particle hydrodynamic velocity from the mass average velocity of the fluid. Following this, we will employ the formalism described here to address the modifications of the convection diffusion migration equation when applied to such finite sized colloidal particles. Before proceeding with the mathematical developments, however, let us physically inspect how a finite sized particle interacts with its surrounding fluid in the purely hydrodynamic limit. For this, one must recognize that particles are rigid bodies, which do not deform like a fluid. Under pure translation, every point on the particle will have the same velocity. When different points in the particle have different velocities, the entire particle will also undergo a rotational motion in addition to translation. The velocity of a point within the particle, U, can be represented as a summation of the translational velocity of the particle center of mass, U0 , and the rotational velocity, , of the point with respect to the center of mass U = U0 +  × (r − r0 )

(6.117)

where r − r0 denotes the position vector of the point relative to the center of mass of the body. When a rigid body (such as a colloidal particle) is placed in a viscous fluid, the no-slip boundary condition will apply at the particle-fluid interface. This means, if we consider a stationary particle in a fluid flowing with a uniform velocity, the fluid in contact with the particle will be stationary. Consequently, the fluid velocity distribution will be altered by the stationary particle. The altered fluid velocity distribution will give rise to a viscous stress, which will be felt by every point on the particle surface. = Integrating the stress tensor, τ , given by Eq. (6.114) over the surface of the particle,

206

FUNDAMENTAL TRANSPORT EQUATIONS

one can determine the total force acting on the particle as   

 = = −p I + µ ∇u + (∇u)T · n dS F = τ · n dS = S

(6.118)

S

where n is the unit outward surface normal vector and S is the surface area of the particle. Similarly, the torque, T, on the particle can be determined from  = (6.119) T = (r − r0 ) × τ · n dS S

Naturally, if the particle suspended in the fluid experiences a force F, an equal and opposing force should be exerted by the particle on the surrounding fluid. In this context, one can determine the modification of the fluid velocity due to the particle by considering the perturbation of the fluid velocity due to a concentrated force F originating at the center of mass of the particle.1 A formal approach for analyzing the fluid velocity field due to a suspended particle involves exploration of the so-called singular solutions of the Stokes creeping flow equations. The basic starting point for such an analysis is to consider the modification of the fluid velocity by a point force, f0 , acting at the origin of a coordinate system. The body force experienced by the fluid due to this point force is represented as a delta distribution, given by fb = δ(r)f0

(6.120)

where fb is the body force per unit volume of the fluid, given by the last term in Eq. (6.3), δ(r) represents a Dirac delta function centered at the origin, and r is the position vector of a point in the fluid. Substituting this force in the creeping flow equations [i.e., neglecting the u · ∇u term in Eq. (6.3)], one obtains the Stokes equation, ∇ ·u=0

(6.121)

0 = −∇p + µ∇ 2 u + δ(r)f0

(6.122)

where the gravity force is also neglected from the Navier–Stokes equation. Equation (6.122) is linear, and hence, it can be shown that the solution of the creeping flow equations will provide a linear dependence of the fluid velocity vector on the point force. In other words, one may relate the fluid velocity vector to the point force through =

u = O · f0

(6.123)

=

where O is the Oseen tensor. It is represented as =

O=

1 = rr

I+ 2 8π µr r

(6.124)

1 Strictly speaking, this is a far field approximation, valid only at sufficiently large distances from the particle.

207

6.3 HYDRODYNAMICS OF COLLOIDAL SYSTEMS

where r = |r| is the distance from the point force. Equation (6.123) represents the velocity distribution of the fluid in presence of a point force at the origin. One can use this basic formalism, and apply it to any distribution of point forces by simply adding up all the contributions due to the individual point forces. Such a linear superposition is applicable in the creeping flow limit where the Stokes equations are linear. To illustrate how the above procedure is applied to a distribution of point forces, let us consider a finite sized particle in the fluid. At each point on the surface of the = particle, there is a force per unit area, f0′ , acting due to the fluid stress, τ · n. One can determine the velocity field of the fluid in presence of such a finite particle by adding up all these individual forces (f0′ ) originating from different points of the surface of the particle. In other words, the velocity of the fluid can be written as  = (6.125) u = O(r − r′ ) · f0′ (r′ ) dS S

where r′ represents the position vectors of the distributed point forces on the surface of the particle. Consider a sphere of radius a moving with a uniform translational velocity, U0 , in an otherwise stationary fluid. The center of the sphere is assumed to be at the origin of the coordinate system. The fluid velocity at an infinite distance from the sphere is zero, while the velocity at the surface of the sphere is U0 . Let us assume that the local velocity of the fluid at every point on the surface of the sphere is caused by a point force that is directly proportional to the particle velocity, i.e., (6.126)

f0′ (r′ ) = cU0

Here c is a constant of proportionality that needs to be determined. The criterion for determination of this proportionality constant is that a proper choice will satisfy the flow boundary conditions. Substituting this force from Eq. (6.126) and the Oseen tensor, Eq. (6.124), in Eq. (6.125), one obtains c u(r) = 8π µ



S


 = 1 (r − r′ )(r − r′ ) I+ · U0 dS |(r − r′ )| |(r − r′ )|2

(6.127)

where S is the surface of the spherical particle, and r′ is the position vector of a point on the sphere surface. Evaluation of the above integral, after substituting a proper choice for the constant c yields u(r) =



 3 a = rr 1  a 3 = rr

I+ 2 + I −3 2 · U0 4r r 4 r r

(6.128)

The above formulation for the velocity field satisfies the boundary conditions u=0 u = U0

as r → ∞ at sphere surface, r = r

(6.129) ′

(6.130)

208

FUNDAMENTAL TRANSPORT EQUATIONS

It is evident from the foregoing discussion that the fluid surrounding the translating spherical particle will acquire an induced velocity due to the particle. The fluid in contact with the particle will move at a velocity U0 , and the velocity distribution will slowly decay to zero as r → ∞ obeying Eq. (6.128). The velocity field in the fluid surrounding the particle will decay as ∼1/r at sufficiently large distances from the particle. This implies that the presence of the particle is felt by the fluid up to a considerable distance. This is the underlying mechanism of hydrodynamic interaction between colloidal entities. The perturbations in the fluid velocity is felt up to large distances from a moving particle, and can, in turn affect the motion of another particle. One can substitute the velocity distribution, Eq. (6.128), in Eq. (6.118) to determine the force exerted on the particle. The subsequent integration leading to the closed form expression for the force is quite laborious. However, the calculations lead to the well-known Stokes expression for the drag force exerted by the particle on the fluid as F = −6π µaU0

(6.131)

EXAMPLE 6.3 Fluid Velocity Field Surrounding a Spherical Particle Translating at a Uniform Velocity. Consider a spherical particle of radius a translating at a uniform velocity, Up , directed along the z axis in an otherwise quiescent fluid. The particle motion is due to a force, F, also acting along the z direction. The situation is depicted in Figure 6.2. Determine the velocity distribution in the fluid surrounding the particle. Solution For the given geometry, the velocity vector Up directed along the z coordinate is first written in the spherical coordinate system as Up = Ur ir + Uθ iθ

Figure 6.2. Translation of a sphere in a quiescent fluid.

209

6.3 HYDRODYNAMICS OF COLLOIDAL SYSTEMS

with Ur = Up cos(θ)

and

Uθ = −Up sin(θ)

where Up = Up · iz . The velocity distribution in the fluid surrounding the particle is given by Eq. (6.128). Using the nomenclature of Figure 6.2, we can recast the velocity distribution as   3 a = rr 1  a 3 = rr

u(r) = I+ 2 + · Up (6.132) I −3 2 4r r 4 r r In spherical coordinates (r, θ, φ), assuming symmetry in φ direction, we have
 = 1 0 I= 0 1 and
 rr 1 0 = 0 0 r2 Substituting these in Eq. (6.132), we obtain 

   3a 1  a 3 1 1 0 1 0 u(r) = + + 0 0 1 0 0 4r 4 r    1 0 Ur −3 · Uθ 0 0

0 1



Upon simplification, the above equation yields

 a 2   a 2  1a 1a u(r) = Ur ir + Uθ iθ 3− 3+ 2r r 4r r

(6.133)



 a 2   a 2  1a 1a u(r) = Up cos(θ)ir − Up sin(θ)iθ 3− 3+ 2r r 4r r

(6.134)

Substituting for Ur and Uθ , the velocity distribution is given by

Equation (6.134) is the required form of the fluid velocity distribution around the translating particle. Let us now see how the Oseen solution is recovered from the above velocity field by rendering the particle infinitesimally small. For the particle translating at a velocity Up along the z direction under the influence of a force of magnitude F , also acting in the z direction, the velocity and and the force are related through the mobility as Up = ωp F =

F 6π µa

(6.135)

where µ is the fluid viscosity and a is the particle radius. This is the well-known Stokes friction factor for a spherical particle. Substituting the above relationship in

210

FUNDAMENTAL TRANSPORT EQUATIONS

Eq. (6.134), one obtains

 a 2  F  a 2  F 1a 1a u(r) = cos(θ)ir − sin(θ)iθ 3− 3+ 2r r 6π µa 4r r 6π µa (6.136) Taking the limit a → 0, the above expression simplifies to u(r) =

1 1 F sin(θ)iθ F cos(θ)ir − 4π µr 8π µr

(6.137)

This is the Oseen form of the fluid velocity field obtained under the influence of a point force F acting at the origin as given in Eqs. (6.123) and (6.124). The procedure for calculating the fluid velocity field around a finite sized colloidal particle delineated above can be applied for other types of flows as well. For instance, one can determine the velocity distribution around a particle subjected to a one-dimensional shear flow. The approach is also extended to study the hydrodynamic interaction between two particles. Providing a detailed analysis of these hydrodynamic interactions is beyond the scope of this book. Several texts discuss hydrodynamic interactions in considerable detail, and the reader is referred to these books for further information (Lamb, 1932; Happel and Brenner, 1965; Russel et al., 1989; Dhont, 1996). It is, however, apparent from the above discussion that a particle moving under the influence of an external force will influence the velocity of the fluid surrounding it. This, in turn, will affect the velocity of another particle when it is sufficiently close to the first particle. The modification of the particle velocity is a function of the separation distance between the particles. In colloidal phenomena of interest in this book, we will encounter such modifications of particle velocities due to hydrodynamic interactions when studying electrophoresis, sedimentation, coagulation, and particle deposition. In all these cases, we will encounter situations when one particle is sufficiently close to another particle, or to a solid surface. In analyzing the hydrodynamic velocities of the particles in close proximity, we will generally represent the particle hydrodynamic velocity as =

vp = f (r) · u

(6.138)

=

where the tensor f (r) represents the hydrodynamic correction functions acting along different coordinate directions. These functions will all have a value of 1 as r → ∞, but will have different values at finite separations between the particles and surfaces. The functions will be different for different flow situations, as well as different geometries. Further details on these functions will be provided in the later chapters as needed. Noting that the hydrodynamic interactions between finite particles can lead to a modification of the particle hydrodynamic velocity from the mass average velocity of the fluid, one can write the particle flux equation in terms of the particle hydrodynamic velocity, vp . Consider the number flux j∗∗ i = ni vi = ni vp + ni (vi − vp )

(6.139)

6.3 HYDRODYNAMICS OF COLLOIDAL SYSTEMS

211

Equation (6.139) is the analogous form of Eq. (6.25), except for the use of the particle hydrodynamic velocity instead of the mass average fluid velocity, u. It should be noted that all the other fluxes of the particle, namely, diffusion and migration fluxes, are determined relative to the particle hydrodynamic velocity for finite sized particles. In other words, the drift velocity produced by a force acting on the finite sized particle is given by vi′ = vi − vp

(6.140)

As a consequence, the diffusion coefficients calculated for finite sized particles differ from the corresponding diffusion coefficients used for point mass solutes. In fact, the hydrodynamic correction functions are also applied to determine the diffusion coefficient of finite sized particles, and the particle diffusivity becomes a function of separation distance between the particles. In general, the particle diffusion coefficient is expressed in terms of a diffusion tensor and is anisotropic. For instance, when a particle is subjected to a shear flow parallel to a stationary planar wall, the diffusivity of the particle at small separations from the wall in the direction parallel to the wall is different from the diffusivity perpendicular to the wall. Denoting the diffusion tensor = for the particle as Di , the diffusive number flux of the particle is given by =

j∗∗ i,D = −Di · ∇ni

(6.141)

where ni is the particle number concentration. Similarly, the migration flux under the influence of an external force, FExt , is given by j∗∗ i,M =

ni = D · FExt kB T

(6.142)

Using the above expressions, the total number flux of particles due to convection, diffusion, and migration in a suspension can be written as =

j∗∗ i = ni vp − Di · ∇ni +

ni = Di · FExt kB T

(6.143)

Similar expressions for the mass and molar fluxes can also be written. One can substitute the particle flux equations in the convection–diffusion–migration Eq. (6.58) and express the spatial particle concentration distribution as
 = ∂ni ni = ∗∗ ∗∗ = −∇ · j∗∗ + R = −∇ · n v − D · ∇n + D · F i p i i Ext + Ri i i ∂t kB T (6.144) At steady state, neglecting any source or sink term (Ri∗∗ = 0), Eq. (6.144) simplifies to 
= ni = Di · FExt = 0 (6.145) ∇ · ni vp − D · ∇ni + kB T

These transport equations for finite size colloidal particles are perfectly analogous to the transport equations for ions (point mass solutes) except for the use of the particle

212

FUNDAMENTAL TRANSPORT EQUATIONS

hydrodynamic velocities and the incorporation of the diffusion tensor containing the hydrodynamic interaction corrections.

6.4 SUMMARY OF GOVERNING EQUATIONS In Chapters 3, 5, and 6, the equations needed to solve electrokinetic problems were derived and discussed. A summary of the governing equations is given below. In the presentation of these equations, the absence of chemical reactions, and constant physical properties (density, viscosity, and dielectric constant) are assumed. Conservation of Mass for an Electrolyte Solution ∇ ·u=0 Conservation of Ionic Species ∂ρi + ∇ · ji = 0 ∂t or ∂ci + ∇ · j∗i = 0 ∂t or ∂ni + ∇ · j∗∗ i =0 ∂t Conservation of Current Density 


∂  ∂(ǫE) +i =0 F zi ci + ∇ · i = ∇ · ∂t ∂t

At steady state, we have ∇ · i = 0.

Momentum Equation (Equation of Motion) ρ

∂u + ρu · ∇u = −∇p + µ∇ 2 u + ρg − ρf ∇ψ ∂t

The creeping flow (low Reynolds number) form is ρ

∂u + ∇p = µ∇ 2 u + ρg − ρf ∇ψ ∂t

where E = −∇ψ and ρf =



zi eni = F



zi ci

6.4 SUMMARY OF GOVERNING EQUATIONS

213

Poisson Equation ǫ∇ 2 ψ = −ρf Poisson–Boltzmann Equation ∇ 2 ψ eq = −

 e zi eψ eq zi ni∞ exp − ǫ i kB T

Ionic Flux: Nernst–Planck Equation zi eρi Di ∇ψ kB T zi Fρi Di = ρi u − Di ∇ρi − ∇ψ RT

ji = ρi vi = ρi u − Di ∇ρi −

in kg/m2 s, zi eci Di ∇ψ kB T zi Fci Di = ci u − Di ∇ci − ∇ψ RT

j∗i = ci vi = ci u − Di ∇ci −

in mol/m2 s, and zi eni Di ∇ψ kB T zi Fni Di = ni u − Di ∇ni − ∇ψ RT

j∗∗ i = ni vi = ni u − Di ∇ni −

in 1/m2 s. Nernst–Einstein Relationship µ∗i =

zi eDi zi FDi = RT kB T

where µ∗i is the mobility of an ion (defined as velocity per unit applied field). Stokes–Einstein Relationship ωi =

1 Di = fi kB T

where ωi is the mobility defined as velocity per unit applied force. fi is the friction factor of an ion, which is given by 6π aµ for spheres of radius a. In some literature it is denoted as λi . For a spherical particle of radius a, D=

kB T 6π aµ

214

FUNDAMENTAL TRANSPORT EQUATIONS

The two mobilities are related by µ∗i = zi eωi Current Density F 2 ∇ψ  2 z i D i ci RT   e2 ∇ψ  2 = eu zi ni − e Di zi ∇ni − z i D i ni kB T

i = Fu



zi ci − F



Di zi ∇ci −

Electrolyte solution electrical conductivity σ is given by σ =

F2  2 e2  2 zi Di ci = zi Di ni RT kB T

Some Useful Relationships F = eNA R = kB NA n i = N A ci zi F zi e = kB T RT Particle Hydrodynamic Velocity =

vp = f (r) · u =

where the tensor function f represents the hydrodynamic corrections for the particle velocity. Particle Flux Equation In terms of number concentration, ni , the particle flux is represented using the particle hydrodynamic velocity as =

j∗∗ i = ni vp − Di · ∇ni +

ni = Di · FExt kB T

=

where Di is the particle diffusion tensor and FExt is the external force. Equations for Continuity and Momentum Conservation Tables 6.6 to 6.9 provide the governing equations for continuity (mass conservation) and the governing equations for momentum conservation in Cartesian, cylindrical, and spherical coordinate systems for constant physical properties.

6.4 SUMMARY OF GOVERNING EQUATIONS

215

TABLE 6.6. Continuity (Conservation) Equation in Several Coordinate Systems for Constant Fluid Density: ∇ · u = 0. Cartesian coordinates (x, y, z): ∂ ∂ ∂ (ux ) + (uy ) + (uz ) = 0 ∂x ∂y ∂z Cylindrical coordinates (r, θ, z): 1 ∂ 1 ∂ ∂ (rur ) + (uθ ) + (uz ) = 0 r ∂r r ∂θ ∂z Spherical coordinates (r, θ, φ): ∂ ∂ 1 1 1 ∂ 2 (r ur ) + (uθ sin θ ) + (uφ ) = 0 r 2 ∂r r sin θ ∂θ r sin θ ∂φ

TABLE 6.7. Equations of Motion in Cartesian Coordinates (x, y, z). In terms of velocity gradient for a Newtonian electrolyte solution with constant ρ, µ, and ǫ: x-component  ∂ux ∂ux ∂ux ∂ux ∂p ρ + ux + uy + uz =− ∂t ∂x ∂y ∂z ∂x  2 ∂ 2 ux ∂ 2 ux ∂ψ ∂ ux + + + ρgx − ρf +µ ∂x 2 ∂y 2 ∂z2 ∂x y-component  ∂uy ∂uy ∂uy ∂uy ∂p ρ + ux + uy + uz =− ∂t ∂x ∂y ∂z ∂y  2 ∂ uy ∂ 2 uy ∂ 2 uy ∂ψ +µ + + + ρgy − ρf 2 2 2 ∂x ∂y ∂z ∂y z-component ρ



∂uz ∂uz ∂uz ∂uz ∂p + ux + uy + uz =− ∂t ∂x ∂y ∂z ∂z  2 2 2 ∂ uz ∂ uz ∂ψ ∂ uz + + + ρgz − ρf +µ ∂x 2 ∂y 2 ∂z2 ∂z

216

FUNDAMENTAL TRANSPORT EQUATIONS

TABLE 6.8. Equations of Motion in Cylindrical Coordinates (r, θ, z). In terms of velocity gradient for a Newtonian electrolyte solution with constant ρ, µ, and ǫ: r-component  ∂ur ∂ur uθ ∂ur u2 ∂ur ∂p ρ + ur + − θ + uz =− ∂t ∂r r ∂θ r ∂z ∂r 
 2 1 ∂ ur 2 ∂uθ ∂ 2 ur ∂ψ ∂ 1 ∂ (rur ) + 2 − + + ρgr − ρf +µ ∂r r ∂r r ∂θ 2 r 2 ∂θ ∂z2 ∂r θ-component  ∂uθ ∂uθ uθ ∂uθ ur uθ ∂uθ 1 ∂p ρ + ur + + + uz =− ∂t ∂r r ∂θ r ∂z r ∂θ 
 2 ∂ 2 uθ ∂ 1 ∂ 1 ∂ uθ 2 ∂ur 1 ∂ψ (ruθ ) + 2 + + +µ + ρgθ − ρf ∂r r ∂r r ∂θ 2 r 2 ∂θ ∂z2 r ∂θ z-component  ∂uz uθ ∂uz ∂uz ∂p ∂uz + ur + + uz =− ρ ∂t ∂r r ∂θ ∂z ∂z  
2 2 1 ∂ ∂ uz ∂ψ ∂ 1 ∂ uz + + ρgz − ρf r uz + 2 +µ r ∂r ∂r r ∂θ 2 ∂z2 ∂z

TABLE 6.9. Equations of Motion in Spherical Coordinates (r, θ, φ). In terms of velocity gradient for a Newtonian electrolyte solution with constant ρ, µ, and ǫ: r-component   u2θ + u2φ ∂ur ∂ur uθ ∂ur uφ ∂ur ∂p + ur + + − ρ =− ∂t ∂r r ∂θ r sin θ ∂φ r ∂r 
2 ∂ψ 2 2 ∂uθ 2 ∂uφ − 2 uθ cot θ − 2 + µ ∇ 2 ur − 2 ur − 2 + ρgr − ρf r r ∂θ r r sin θ ∂φ ∂r θ-component   u2φ cot θ uθ ∂uθ uφ ∂uθ ur uθ 1 ∂p ∂uθ ∂uθ ρ + ur + + + − =− ∂t ∂r r ∂θ r sin θ ∂φ r r r ∂θ 
2 cos θ ∂uφ uθ 1 ∂ψ 2 ∂ur − 2 2 − 2 2 + ρgθ − ρf + µ ∇ 2 uθ + 2 r ∂θ r ∂θ r sin θ r sin θ ∂φ (continued)

6.5 NOMENCLATURE

TABLE 6.9. Continued. φ-component  ∂uφ uθ ∂uφ uφ ∂uφ uφ ur u φ uθ ∂uφ + ur + + + + cot θ = ρ ∂t ∂r r ∂θ r sin θ ∂φ r r
 2 ∂u 2 cos θ ∂uθ u 1 ∂p φ r + µ ∇ 2 uφ − 2 2 + 2 + 2 2 − r sin θ ∂φ r sin θ ∂φ r sin θ r sin θ ∂φ + ρgφ − ρf

1 ∂ψ r sin θ ∂φ

In these equations, ∇2 =

1 ∂ r 2 ∂r



r2

∂ ∂r



+

∂ 1 r 2 sin θ ∂θ



sin θ

∂ ∂θ



+

1 r 2 sin2 θ



∂2 ∂φ 2



6.5 NOMENCLATURE a c ci cs C Cp D Di = Di e E F F fEl FExt f0 fb fE fi = f (r) g = I i ix ji j∗i j∗∗ i

radius of a spherical particle, m total molar concentration, mol/m3 molar concentration of the i th species, mol/m3 molar concentration of solvent, mol/m3 scaled ionic concentration ion/m3 fluid specific heat, J/kg K diffusion coefficient, m2 /s diffusion coefficient of the i th species, m2 /s diffusion tensor elementary charge, C electric field strength vector, V/m Faraday number, C/mol force acting on a body due to fluid stress, N electrical force, N external force, N point force acting at origin body force per unit volume of a fluid element, N/m3 electric body force, N/m3 Stokes–Einstein friction factor tensor describing the particle hydrodynamic correction functions gravitational acceleration vector, m/s2 unit tensor current density vector, A/m2 current density in the x-direction, A/m2 mass flux vector of ion i relative to a stationary observer, kg/m2 s molar flux vector of ion i relative to a stationary observer, mol/m2 s ionic flux vector of ion i relative to a stationary observer, 1/m2 s

217

218

kB kf M M Mi NA n ni ni∞ = O p R r t T T u ui u∗ vi vi′ vp vs wi (x, y, z) xi , yi z zi

FUNDAMENTAL TRANSPORT EQUATIONS

Boltzmann constant, J/K fluid thermal conductivity, W/mK solution molarity, mol/dm3 average molar mass, kg/kmol molar mass of ion i, kg/kmol Avogadro number, mol−1 unit outward normal to a surface number concentration of ion i, m−3 number concentration of ion i in the bulk solution, m−3 Oseen tensor fluid pressure, Pa or N/m2 gas constant, J/mol K position vector of a point, m time, s temperature, K torque on a rigid body, Nm mass local average velocity vector relative to a stationary observer, m/s velocity vector of species i relative to a stationary observer, m/s molar average velocity vector relative to a stationary observer, m/s velocity of species i, m/s drift velocity (vi − u), m/s particle hydrodynamic velocity, m/s solvent velocity, m/s mass fraction, dimensionless Cartesian coordinates, m mole fraction, dimensionless absolute value of the valence for a (z : z) electrolyte valence of the i th ionic species

Greek Symbols ǫ λi µ µi µ∗i ρ ρf ρi ρs σ σM ◦ σM

permittivity of a material, C/mV molar electric conductivity of the i th ionic species, Sm2 /mol fluid viscosity, kg/ms or Pa s electrochemical potential of the i th species, J/mol mobility of i th charge carrier species, m2V−1 s−1 fluid density, kg/m3 free charge density, C/m3 mass density of the i th species, kg/m3 mass density (concentration) of the solvent, kg/m3 electrical conductivity of the electrolyte solution, S/m or A/Vm molar electric conductivity of an electrolyte, Sm2 /mol molar electric conductivity of an electrolyte at infinite dilution, Sm2 /mol

6.7 REFERENCES =

τ ψ ψ eq ψ∞ ωi

∇ ∇2

219

hydrodynamic stress tensor, N/m2 total electric potential, V electric potential in Poisson–Boltzmann equation for stationary system, V potential in the bulk electrolyte solution, V mobility of i th species, mN−1 s−1 angular velocity of a point, rad/s del operator, m−1 Laplacian operator, m−2

Subscripts i x y z

i th ionic species x-direction y-direction z-direction

6.6 PROBLEMS 6.1. Evaluate the electrolyte solution electric conductivity and its molar conductivity for a 0.5 mM BaCl2 solution at 25◦ C using the equivalent ionic conductivities at infinite dilution provided in Table 6.3. Compare your results with the data provided in Table 6.2. 6.2. Assuming infinite dilution, evaluate the solution conductivity and its molar conductivity for a 0.3 g/L Na2 SO4 solution at 25◦ C using the equivalent ionic conductivities provided in Table 6.3. Compare your results with the data in Table 6.2. 6.3. A volume of 0.5 L of 0.001 M BaCl2 is added to 0.3 L of 0.0005 M KCl at 25◦ C. What is the final electric conductivity of the solution? Now, let us add 0.4 L of de-ionized water to the mixed salt solution. What is the electric conductivity of the diluted solution? 6.4. An electrolyte solution fills a 0.2 m long circular tube. The tube ends are placed between two electrodes having an electric potential difference of 150 V. The electrolyte solution contains 1.2 mM CaCl2 . What is the electric current density? Assume a temperature of 25◦ C and an infinitely dilute solution.

6.7 REFERENCES Aris, R., Vectors, Tensors, and Basic Equations of Fluid Mechanics, Dover, New York, (1989). Atkinson, G., Electrochemical Information, in American Institute of Physics Handbook, 3rd ed. Gray, D. E. (Ed.), McGraw-Hill, New York, (1972).

220

FUNDAMENTAL TRANSPORT EQUATIONS

Bird, R. B., Stewart, W. E., and Lightfoot, E. N., Transport Phenomena, Wiley, New York, (1960, 2002). Dhont, J. K. G., An Introduction to the Dynamics of Colloids, Elsevier, Amsterdam, (1996). Einstein, A., Investigations on the Theory of Brownian Motion, Dover Publications, New York, (1956). Happel, J., and Brenner, H., Low Reynolds Number Hydrodynamics, Prentice-Hall, NJ (1965). Lamb, H., Hydrodynamics, Cambridge University Press, (1932). Marsh, K. N., (Ed.), Recommended Reference Materials for the Realization of Physicochemical Properties, Blackwell, Oxford, (1987). Moon, P. H., and Spencer, D. E., Field Theory Handbook: Including Coordinate Systems, Differential Equations and Their Solutions, 2nd ed., Springer-Verlag, New York, (1971). Newman, J. S., Electrochemical Systems, 2nd ed., Prentice Hall, Englewood Cliffs, New Jersey, (1991). Probstein, R. F., Physicochemical Hydrodynamics, An Introduction, 2nd ed., Wiley Interscience, New York, (2003). Russel, W. B., Saville, D. A., and Schowalter, W. R., Colloidal Dispersions, Cambridge Univ. Pr., Cambridge, (1989). Schey, H. M., Div, Grad, Curl and All That, 3rd ed., Norton, New York, (1997). Verbrugge, M. W., and Pintauro, P. N., Transport Models for Ion-Exchange Membranes, Modern Aspects of Electrochemistry, No. 19, Conway, B. E., Bockris, J. O’M., and White, R. E., (Eds.), Plenum, New York, (1989). Wu, Y. C., Pratt, K. W., and Koch, W. F., Determination of the absolute specific conductance of primary standard KCl solutions, J. Solution Chem., 18, 515–528, (1989).

CHAPTER 7

ELECTROKINETIC PHENOMENA

Electrokinetics is a general term associated with the relative motion between two charged phases. Electrokinetic phenomena occur when one attempts to shear off the mobile part of the electric double layer. Then, as the charged surface (plus attached material) tends to move in the appropriate direction, the ions in the mobile part of the electric double layer undergo a net migration in the opposite direction, carrying solvent along with them, thereby causing the movement of the solvent. Similarly, an electric field is created if the charged surface and the diffuse part of the double layer are made to move relative to each other (Shaw, 1980; Hiemenz and Rajagopalan, 1997). Among the many types of phenomena that might occur as a result of relative motion between charged phases and electrolytes, four types of electrokinetic phenomena are more commonly encountered: electroosmosis, streaming potential, electrophoresis, and sedimentation potential. These four types of electrokinetic phenomena are briefly described below. 7.1 ELECTROOSMOSIS Electroosmosis represents the movement, due to an applied electric field, of an electrolyte solution relative to a stationary charged surface (i.e., a capillary tube or porous media). The pressure necessary to counterbalance electroosmotic flow is termed the electroosmotic pressure. A typical electroosmotic fluid flow in a capillary tube is shown in Figure 7.1. When the capillary tube is negatively charged, the applied electric field exerts a force in the Electrokinetic and Colloid Transport Phenomena, by Jacob H. Masliyah and Subir Bhattacharjee Copyright © 2006 John Wiley & Sons, Inc.

221

222

ELECTROKINETIC PHENOMENA

Figure 7.1. Electroosmotic flow in a capillary tube.

direction of the cathode on the excess ions of positive charge near the surface. The positively charged ions then drag the electrolyte solution along with them and cause the fluid to flow towards the cathode. Note that one should impose the experimental condition that the pressure difference between the two capillary ends be zero to study electroosmosis under the influence of an applied electric field. Electroosmosis can be employed to drain porous media and in the evaluation of the surface charge of capillary tubes or porous media (see, e.g., Hiemenz and Rajagopalan, 1997; Probstein, 2003). The electroosmotic pressure between the two ends of the capillary can be measured when there is no flow through the capillary under the influence of the applied electric field. 7.2 STREAMING POTENTIAL An electric field is created when an electrolyte solution is made to flow along a stationary charged surface by applying a pressure gradient. Such flows are generally encountered in narrow capillary microchannels connected to two reservoirs. When the electrolyte concentrations in the two reservoirs are identical, and when there is no net current flowing through the system, the steady-state electric field developed between the two reservoirs is called the streaming potential. For example, a streaming potential is set up when an electrolyte solution is pumped through a negatively charged capillary as shown in Figure 7.2. The electric field due to the flow is from right to left. The principle of streaming potential is used in sea water desalination. The streaming potential phenomenon is considered as a reciprocal (i.e., opposite) of the phenomenon of electroosmosis. 7.3 ELECTROPHORESIS The movement of a charged surface, such as that of a colloidal particle, relative to a stationary liquid caused by an applied electric field is known as electrophoresis. A typical particle electrophoresis is shown in Figure 7.3. Due to the presence of the anode and cathode terminals, an electric field, E, becomes established from left to

7.5 NON-EQUILIBRIUM PROCESSES AND ONSAGER RELATIONSHIPS

223

Figure 7.2. Development of streaming potential when an electrolyte is pumped through a capillary.

Figure 7.3. Electrophoresis of a charged particle in an external electric field.

right. Under the influence of this electric field, the negatively charged colloidal particle migrates towards the anode. Electrophoresis is usually employed in measuring the surface potential of a charged particle. Note that in electrophoresis, one does not apply any pressure gradients to cause a flow. 7.4 SEDIMENTATION POTENTIAL An electric field is created when charged particles move relative to a liquid. The movement of the particles can be under gravitational or centrifugal fields. This phenomenon is sometimes called the Dorn effect or the migration potential. It is the least studied among the electrokinetic phenomena. The sedimentation potential of a settling suspension of charged colloidal particles under the influence of a gravitational field is illustrated in Figure 7.4. In a strict sense, the sedimentation potential is defined for the case when the current flow is zero in such processes. 7.5 NON-EQUILIBRIUM PROCESSES AND ONSAGER RELATIONSHIPS In electrokinetic processes, one is concerned with the coupled influence of multiple types of forces or potentials (electrical, pressure, gravity, etc.) on the transport

224

ELECTROKINETIC PHENOMENA

Figure 7.4. Sedimentation of charged colloidal particles under gravity setting up a sedimentation potential.

behavior of a multicomponent system. These types of processes are often described in light of non-equilibrium thermodynamic theories by means of relationships known as the Onsager reciprocity relations (Onsager, 1931a,b; de Groot and Mazur, 1962). A brief outline of the pertinent information on non-equilibrium thermodynamics and the Onsager relationships is presented here. Several physical phenomena encountered in natural and engineered systems are governed by transport processes where a simple linear relationship between a flux and the corresponding driving force exists. The driving force is generally a gradient of some potential such as concentration, chemical potential, temperature, pressure, electric potential, etc. Some examples of transport processes governed by such a simple proportionality between a flux and its conjugate driving force are: (i) molecular diffusion, where the diffusive flux is related to the concentration gradient, (ii) thermal conduction, where the heat flux is related to the temperature gradient, (iii) fluid flow, where the flow rate is proportional to the pressure gradient, and (iv) electrical conduction, where the current is related to the applied potential gradient. The equations formulated to describe such proportionality are often called constitutive laws. For instance, the diffusive flux is related to the concentration gradient through Fick’s law, Fourier’s law relates the heat flux to the temperature gradient in heat conduction, Newton’s law (manifested as Navier–Stokes or Darcy’s equation) governs the relationship between fluid flow and pressure gradient, and finally, Ohm’s law relates the current to the electric potential gradient during electrical conduction. In all the above cases, the constitutive law states that the flux is directly proportional to the conjugated driving force. In other words, one can state Jα = Lαα Xα

(7.1)

where Jα is the flux, Xα is the conjugate driving force, and Lαα is the proportionality constant or the transport coefficient. These transport processes are termed non-equilibrium thermodynamic phenomena, since the existence of the driving force in these processes signifies a deviation from equilibrium.

7.5 NON-EQUILIBRIUM PROCESSES AND ONSAGER RELATIONSHIPS

225

The electrokinetic transport processes introduced in this chapter, and further discussed in the subsequent chapters, can not be described in terms of a simple constitutive relationship between a single driving force and the corresponding flux. For instance, when an electric potential gradient is applied along the axis of a capillary containing an electrolyte solution, not only does a current flow occur, but there is also a coupled flow of the liquid along the capillary. In this case, one generally states that the electric potential gradient gives rise to a non-conjugated flow of the liquid, implying that flow is not related to the electric potential gradient through a direct constitutive equation. Conversely, when a pressure gradient is applied across the capillary, not only will this pressure drop cause a fluid flow, but also a coupled flow of current. A similar type of correspondence exists between electrophoresis and sedimentation. In the above instances, we observe that there exists a coupling between driving forces of one type and fluxes of another type. For sufficiently slow processes, any flow (or flux) may depend in a linear manner not only on its conjugate force, but other nonconjugate forces. The study of such coupled transport phenomena falls under the purview of non-equilibrium thermodynamics. Mathematically, the nonconjugate or cross effects of different forces on the fluxes are described by adding new terms to the constitutive equations of the form given by Eq. (7.1). The modified constitutive equations are written in a general form relating each type of flux with all the driving forces as (Onsager, 1931a,b; de Groot and Mazur, 1962) J1 = L11 X1 + L12 X2 + L13 X3 + · · · + L1n Xn J2 = L21 X1 + L22 X2 + L23 X3 + · · · + L2n Xn (7.2)

··· = ··· ··· = ··· Jn = Ln1 X1 + Ln2 X2 + Ln3 X3 + · · · + Lnn Xn which can be written in an abbreviated form as Jα =

n


Lαβ Xβ

α, β = 1, 2, . . . , n

(7.3)

β=1

where Jα stands for different fluxes, Xβ represents different driving forces, and Lαβ represents the phenomenological coefficients, which are independent of both the fluxes and the driving forces. The term phenomenological coefficients signifies that these coefficients are usually determined from experiments. Although we have used vector notations for the fluxes and driving forces, it should be noted that the fluxes can be scalars or even higher order tensors. Also, the driving forces may not strictly be regarded as forces in a Newtonian sense. These can represent any type of potential gradient causing an irreversible transport phenomenon. The coupled irreversible transport phenomena described by Eq. (7.3) are governed by Onsager’s law, which states that for a proper choice of the fluxes and driving forces,

226

ELECTROKINETIC PHENOMENA

the matrix of the phenomenological coefficients Lαβ is symmetric. In other words Lαβ = Lβα

(7.4)

These identities are referred to as Onsager reciprocal relations (Onsager, 1931a,b). Let us now illustrate the above formalism for electrokinetic flow. In such processes, the current, I , and the volumetric fluid flow rate, Q, are two relevant fluxes, while the electric potential gradient (E = −∇ψ), and the pressure gradient, −∇p, constitute the two corresponding associated driving forces. The transport phenomena can then be written in terms of the coupled transport equations as follows I = L11 E + L12 (−∇p)

(7.5)

Q = L21 E + L22 (−∇p)

(7.6)

Note that in writing the above relationships, the diagonal terms of the coefficient matrix (i.e., the Lαα terms) were chosen such that these phenomenological coefficients relate the fluxes to their “natural” conjugate driving forces. Clearly, for the current, the natural conjugate driving force is the potential gradient, while for the flow rate, the conjugate driving force is the pressure gradient. According to the Onsager reciprocal relations, in this case, we have L12 = L21 . The choice of the natural conjugate driving force for a given flux is non trivial, particularly when dealing with more than two coupled processes (de Groot and Mazur, 1962). In the next chapter, this type of relationship will be used to check the validity of the mathematical formulation for streaming potential and electroosmotic flow. The use of reciprocity relationships allows the consideration of one type of electrokinetic transport process (for instance, electrophoresis) and the application of the results to describe the coupled transport process (i.e., sedimentation) through the use of the appropriate Onsager relationships. 7.6 NOMENCLATURE E E I Jα Lαβ Q Xβ ∇p

magnitude of electric field vector, V/m electric field, V/m current, A flux phenomenological coefficient volumetric flow rate, m3 driving force pressure gradient, Pa/m

7.7 REFERENCES de Groot, S. R., and Mazur, P., Nonequilibrium Thermodynamics, North-Holland, Amsterdam, (1962).

7.7 REFERENCES

227

Hiemenz, P. C., and Rajagopalan, R., Principles of Colloid and Surface Chemistry, 3rd ed., Marcel Dekker, New York, (1997). Onsager, L., Reciprocal relations in irreversible processes I, Phys. Rev., 37, 405–426, (1931a). Onsager, L., Reciprocal relations in irreversible processes II, Phys. Rev., 38, 2265–2279, (1931b). Probstein, R. F., Physicochemical Hydrodynamics, An Introduction, 2nd ed., Wiley Interscience, New York, (2003). Shaw, D. J., Introduction to Colloid and Surface Chemistry, 3rd ed., Butterworths, London, (1980).

CHAPTER 8

FLOW IN MICROCHANNELS

8.1 LIQUID FLOW IN CHANNELS Traditionally, Newtonian liquids are made to flow in channels, pipes, and porous media under the influence of a pressure gradient. When the channel dimension or the pore size is relatively large, such a method for liquid flow is quite effective. It is also possible to affect liquid flow by imposing an electric potential gradient along the flow channel or across a porous bed. Such a flow mode is possible when the channel walls carry a surface charge and the flowing liquid contains free charges. The effectiveness of a particular mode for liquid flow is dependent on the system geometry and the physical properties of the flowing liquid. In this chapter, we will refer to a liquid as being either an electrolyte solution, e.g., water with dissolved ionic species, or a liquid having free charge. A pressure driven laminar flow in a straight channel is normally termed as Poiseuille flow. An electric potential driven flow is termed as electroosmotic flow. Electroosmosis is associated with the movement of a bulk electrolyte solution or a liquid carrying a free charge, relative to a stationary charged surface, under the influence of an imposed electric field. For example, an electrolyte solution in a porous medium can be made to move when a electric field is applied. In many devices, glass capillaries (microchannels) are used for electroosmotic flows. The charge on the glass capillary wall arises from the dissociation of surface silanol groups, −SiOH, or the preferential adsorption of OH− ions onto the glass surface. In most cases, the surface charge is negative and an electric double layer Electrokinetic and Colloid Transport Phenomena, by Jacob H. Masliyah and Subir Bhattacharjee Copyright © 2006 John Wiley & Sons, Inc.

229

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FLOW IN MICROCHANNELS

is present adjacent to the capillary walls. The excess charge, i.e., the absence of electroneutrality within the electric double layer, is responsible for establishing the electroosmotic flow within the capillary under an external electric field. 8.2 ELECTROOSMOTIC FLOW IN A SLIT CHARGED MICROCHANNEL 8.2.1

Electric Potential

In order to understand pressure driven and electroosmotic flow in porous media, in capillary tubes of varying cross-section, or along surfaces with non-homogeneous surface charge distribution, it is first necessary to be able to analyze the simple geometry of the flow in a straight gap formed by two parallel plates. Such a flow configuration is referred to as the slit microchannel. In this analysis, we shall present pressure driven and electroosmotic flow in the gap between two parallel surfaces having a low surface electric potential. The flow to be considered is steady-state and fully developed. In other words, there is no time dependence and there are no end effects. The analysis is that of a one-dimensional flow. By and large, the analysis is that due to Burgreen and Nakache (1964), although their analysis did not invoke the low surface potential assumption. Consider a microchannel formed by two parallel plates, i.e., a slit microchannel. The walls of the channel have a surface electric potential of ψs . The wall potential is assumed to be negative to facilitate plotting the results and subsequent discussions. It should be noted that assigning a negative sign for the wall potential does not impose any restriction on the analysis. Figures 8.1 and 8.2 depict the flow geometry. The microchannel ends are subjected to an electric potential that gives rise to a uniform electric field strength of Ex where Ex = −

∂φ(x, y) ∂x

(8.1)

Figure 8.1. Electroosmotic and pressure driven flow in a microchannel gap formed by two parallel plates. The microchannel walls are negatively charged. The flow is purely electroosmotic when there is an electric field, Ex , acting between the electrodes along the axial direction and P1 = P2 . The flow is pressure driven when P1
= P2 and Ex = 0.

8.2 ELECTROOSMOTIC FLOW IN A SLIT CHARGED MICROCHANNEL

231

Figure 8.2. Electroosmotic flow in a microchannel gap formed by two parallel plates. Details of the two-dimensional slit microchannel geometry are shown. The walls of the microchannel have a surface potential ψs , which sets up an excess counterion concentration near the walls. The shaded regions near the walls represent the electric double layers where an excess counterion concentration exists.

with φ(x, y) being the local electric potential in the microchannel, x being the axial direction, and y being the coordinate normal to the axial direction, i.e., the transverse direction. As discussed earlier, in the microchannel, the concentration of the counterions (in the case of a negatively charged surface, they are the positive ions or simply the cations) predominate near the charged microchannel walls, within the so called Debye sheath. This gives rise to a finite (non-zero) volumetric free charge density, ρf , in the fluid adjacent to the charged walls. The application of an external electric field on this charged fluid results in a net electrical body force, ρf Ex , acting on the fluid close to the microchannel surface. This electric body force results in a fluid flow through the microchannel. Such an electrically driven flow is generally referred to as electroosmotic flow. Consider a symmetric (z : z) electrolyte solution contained in two reservoirs connected by the microchannel. Let the microchannel walls have a surface potential ψs . In this analysis, we will not distinguish between the surface potential and the zeta potential of the microchannel surface. The electric potential at a location (x, y) in the channel, given by φ(x, y), arises due to the superposition of the applied external electric potential and the potential due to the surface charge of the microchannel walls. In other words, the electric potential at a given location is the algebraic sum of the potential due to the electric double layer and the potential due to the imposed electric field. Assuming that the potential due to the electrical double layer is independent of axial position in the microchannel (which is valid for long microchannels, neglecting any end effects), one can write φ(x, y) ≡ φ = ψ(y) + [φ0 − xEx ]

(8.2)

where ψ(y) is the electric potential due to the electric double layer at the equilibrium state corresponding to no fluid motion and no applied external electric field, φ0 is the

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FLOW IN MICROCHANNELS

value of the imposed potential at x = 0, φ0 − xEx is the electric potential at a given axial location due to the electric field strength Ex in the absence of the electric double layer (i.e., zero surface potential), Ex is the electric field strength being independent of position, x is the axial direction, and y is the transverse coordinate of the flow gap. The Poisson equation defining the electric potential within the microchannel in Cartesian coordinates is given by ρf ∂ 2φ ∂ 2φ + =− ∂x 2 ∂y 2 ǫ

(8.3)

Introducing the expression of the electric potential given by Eq. (8.2) into the Poisson equation, Eq. (8.3), gives ρf d 2ψ =− (8.4) 2 dy ǫ Equation (8.4) is the resulting Poisson equation for the slit microchannel. The free charge density, ρf , is given by ρf =




(8.5)

ezk nk

k

Now assume that the ionic number concentration, nk , of the k th ion is given by the Boltzmann distribution. It follows that for a (z : z) electrolyte   zk eψ nk = n∞ exp − (8.6) kB T Here, n∞ is the ionic number concentration in the neutral electrolyte. For a symmetric electrolyte one can write and Eq. (8.6) provides

and

z+ = −z− = z   zeψ n+ = n∞ exp − kB T 

zeψ n− = n∞ exp kB T



(8.7)

(8.8)

The free charge density, Eq. (8.5), can be written as ρf = e(z+ n+ + z− n− ) = ze(n+ − n− ) Making use of the Boltzmann expression, Eq. (8.9) becomes      zeψ zeψ ρf = zen∞ exp − − exp kB T kB T

(8.9)

(8.10)

8.2 ELECTROOSMOTIC FLOW IN A SLIT CHARGED MICROCHANNEL

233

or 



(8.11)

  zeψ 2zen∞ d 2ψ = sinh dy 2 ǫ kB T

(8.12)

zeψ ρf = −2zen∞ sinh kB T

In deriving Eq. (8.11), it was assumed that the local distribution of the free charges (ions) is governed solely by the electrical potential distribution due to the charged wall, namely ψ(y), and not the total potential, φ(x, y). This assumption is made primarily for the sake of convenience, and should be valid if the axial variation of the electric potential is much smaller than the transverse variation of the potential. This condition will generally hold when the microchannel length is much larger than the gap, L ≫ h, where L is the channel length. Combining Eqs. (8.4) and (8.11) leads to

Equation (8.12) is the Poisson–Boltzmann equation that defines the electric double layer potential, ψ(y), in the microchannel. Its derivation was made with the explicit assumption of the validity of the Boltzmann distribution in a flow system. The analytical, albeit fairly complicated, solution of Eq. (8.12) was provided by Burgreen and Nakache (1964). To simplify the analysis, we will assume that zeψ/kB T < 1 so that we can write sinh(zeψ/kB T ) ≈ zeψ/kB T . Such a linearization is known as the Debye–Hückel approximation, as discussed in Chapter 5. For z = 1 and ψ = 0.025 V, the term (zeψ/kB T ) is close to unity. Physically, such an approximation means that the electric potential is small compared to the thermal potential, i.e., |zeψ| < kB T . Therefore, for small potentials, the Poisson–Boltzmann equation (8.12), becomes d 2ψ 2n∞ z2 e2 ψ = dy 2 ǫkB T

(8.13)

Defining the inverse double layer thickness (or inverse Debye length), κ, by κ=



2n∞ z2 e2 ǫkB T

1/2

(8.14)

Eq. (8.13) becomes d 2ψ = κ 2ψ dy 2

(8.15)

κ is also referred to as the Debye–Hückel parameter. Equation (8.15) is a linear differential equation subject to the following boundary conditions: At y = h At y = 0

ψ = ζ (Potential at the shear plane) dψ = 0 (Axis of symmetry) dy

(8.16-a) (8.16-b)

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FLOW IN MICROCHANNELS

Here, h is the half-width gap of the microchannel. The second boundary condition implicitly implies that at y = ±h, ψ = ζ . In other words, the confining microchannel walls carry the same surface potential and one has symmetry within the channel. It should be noted here that the potential at y = h is the zeta potential (ζ ) instead of the actual surface potential of the microchannel wall, ψs . This adjustment is necessary to account for the no-slip boundary condition for the fluid flow problem at the shear plane (cf., Section 5.8), which is discussed below. At this stage, it should be borne in mind that the true surface potential of the wall does not appear in the theory for electrokinetic flows. Instead, the potential at the shear plane, the zeta potential, is used in the theory. Accordingly, the location y = ±h corresponds to the distance between the centerline of the channel and the shear plane, not the channel wall itself. The solution to Eq. (8.15) subject to the boundary conditions of Eqs. (8.16-a) and (8.16-b) gives cosh(κy) ψ =ζ (8.17) cosh(κh) Making use of Eqs. (8.2) and (8.17), the total electric potential is then given by φ(x, y) = ζ

cosh(κy) + [φ0 − xEx ] cosh(κh)

(8.18)

and the free charge density for low surface potential is given by ρf = −ǫκ 2 ψ = −ǫκ 2 ζ

cosh(κy) cosh(κh)

(8.19)

At this point, it is pertinent to discuss the implications of using the Debye– Hückel approximation in the subsequent theoretical developments. The linearization of the Poisson–Boltzmann equation for a symmetric (z : z) electrolyte, Eq. (8.12), involves writing the sinh term on its right hand side as a Taylor series expansion in  = zeψ/kB T . The expansion gives sinh() =  +

3 + ··· 3!

Note that Eq. (8.15) was obtained by retaining only the first term in the above series expansion for sinh(). This implies that the linearized form can be considered as accurate up to O( 2 ), since the next neglected term in the expansion is of O( 3 ). In the subsequent analysis, there will be at least two places where this approximation will have a bearing. In the next subsection, when considering the influence of the electrical body force, ρf Ex , on the fluid velocity distribution in the microchannel, the charge density, ρf , will be written in terms of the linearized form, Eq. (8.19), instead of the general form given by Eq. (8.11). Subsequently, during determination of the migration current density using the Nernst–Planck equations, we will encounter an expression of the form n+ + n− = n∞ [exp(−) + exp()] = 2n∞ cosh()

(8.20)

8.2 ELECTROOSMOTIC FLOW IN A SLIT CHARGED MICROCHANNEL

235

This expression uses the complete Boltzmann distribution. One needs to bear in mind at this step that substituting the potential distribution, Eq. (8.17), obtained from the solution of the linearized Poisson–Boltzmann equation directly in Eq. (8.20) will be mathematically inconsistent. This is because we are mixing a result that is based on at best a quadratic approximation of the Boltzmann distribution with another that is based on the complete Boltzmann distribution. Consequently, one first needs to expand the cosh term in Eq. (8.20) in a Taylor series, where one can retain up to the quadratic term 2 cosh() = 1 + + O( 4 ) 2! If one substitutes Eq. (8.17) in the above Taylor expansion, the resulting expression will be accurate up to O( 2 ) and will be mathematically consistent. Such a check of consistency in the entire mathematical formulation will be extremely important in correct development of the analytical models for electrokinetic flow. The above discussion also implies that the subsequent theoretical development will be somewhat approximate owing to the use of the linearized Poisson–Boltzmann equation. 8.2.2

Flow Velocity

To this point we have solved for the electric parameters by combining the Poisson and Boltzmann equations and invoking the Debye–Hückel approximation. In order to evaluate the axial velocity, ux , we need to consider the modified Navier– Stokes equation where the electric body force is included. For constant fluid properties under laminar flow conditions, the modified Navier–Stokes equation in the x-direction is given by [see Eq. (6.112)]: µ

∂φ ∂p d 2 ux = + ρf − ρgx 2 dy ∂x ∂x

(8.21)

where ux is the axial liquid velocity, µ is the liquid viscosity, ρ is the liquid mass density, gx is the gravitational acceleration in the x-direction at location y, and p is the liquid pressure. The continuity equation for a one-dimensional flow provides ∂ux =0 ∂x

(8.22)

As the flow is fully developed, the velocity in y-direction, uy , is taken as zero and the velocity ux is solely a function of the transverse direction (y). The latter fact is reflected by the solution of the continuity equation, (8.22), which provides ux = f (y)

(8.23)

As the velocity in the transverse direction, uy is zero, the modified Navier–Stokes equation for the y-direction can be written as ∂p ∂φ + ρf − ρgy = 0 ∂y ∂y

(8.24)

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FLOW IN MICROCHANNELS

where gy is the gravitational acceleration in the y-direction. For a horizontal microchannel, the variation in the pressure, ∂p/∂y, is due to the hydrostatic head and the electrostatic body force term of Eq. (8.24). As the microchannel gap is normally very small, and gx = 0 for a horizontal channel, one can assume that the liquid pressure is nearly independent of y and that the pressure is solely a function of the axial direction. Consequently, the modified Navier–Stokes equation for a horizontal channel in the x-direction becomes µ

∂φ dp d 2 ux = + ρf 2 dy dx ∂x

(8.25)

One can think of dp/dx as the axial pressure gradient in the microchannel. As we have assumed fully developed flow, dp/dx is a constant. In a normal channel flow, dp/dx is a negative quantity. For convenience, we set px = −

dp dx

(8.26)

Making use of Eqs. (8.1), (8.19), and (8.26), the momentum equation as represented by the modified Navier–Stokes equation, Eq. (8.25) becomes µ

cosh(κy) d 2 ux = −px + ǫκ 2 Ex ζ 2 dy cosh(κh)

(8.27)

The momentum equation is subject to the following boundary conditions: ux = 0

at y = h

(8.28)

and dux =0 dy

at y = 0

(8.29)

The boundary condition, Eq. (8.28), is a statement that at the channel wall (or rather, at the shear plane) the electrolyte solution velocity is zero, i.e., the no-slip condition is imposed. The boundary condition of Eq. (8.29) implies flow symmetry in the y-direction. This is a reasonable assumption as the electric potential was assumed to be symmetric. Solution of Eq. (8.27) subject to the boundary conditions provided by Eqs. (8.28) and (8.29) is given by    y 2  ǫE ζ  h2 p x cosh(κy) x ux (y) = ux = − 1− 1− 2µ h µ cosh(κh)

(8.30)

8.2 ELECTROOSMOTIC FLOW IN A SLIT CHARGED MICROCHANNEL

237

Equation (8.30) gives the variation of the axial velocity across the channel of an electrolyte solution or a liquid carrying free charges. The first term is the liquid velocity due to an imposed pressure gradient, px . It is normally referred to as Poiseuille flow. The velocity due to the imposed pressure gradient has a parabolic profile. For a given pressure gradient, px , and fluid viscosity, µ, the local velocity is proportional to h2 . Clearly, the flow is much affected by the channel width 2h. As the channel width becomes very small, pressure driven flows become impractical as the pumping requirements become prohibitive. The second term of Eq. (8.30) is due to the electroosmotic flow as a consequence of the imposed electrical potential gradient, Ex . Here, the local velocity is proportional to (ǫEx ζ /µ). For given system properties, the local velocity is directly proportional to the imposed electrical potential gradient. The channel width does not appear in the proportionality term (ǫEx ζ /µ). Consequently, electroosmotic flows become very attractive for very narrow channels. Equation (8.30) indicates that the local velocity due to electroosmotic flow is much dependent on the electric double layer thickness, κ −1 . Figure 8.3 shows the variation of the dimensionless local velocity due to a pressure driven flow. The normalized local velocity due to a pressure driven flow is given as  y 2 ux,press  = Ux,press = 1 − h h2 px /2µ

(8.31)

The normalized local velocity is parabolic when plotted against normalized transverse position, y/ h, as shown in Figure 8.3.

Figure 8.3. Variation of dimensionless local velocity due to a pressure driven flow with normalized transverse position in a slit microchannel.

238

FLOW IN MICROCHANNELS

Figure 8.4. Variation of dimensionless local velocity due to electroosmotic flow with normalized transverse position in a slit microchannel.

Figure 8.4 shows the variation of the dimensionless local velocity due to electroosmotic flow, written for convenience as cosh(κh · yh ) ux,el ux,el = = Ux,el = 1 − − (ǫEx ζ /µ) − Ex cosh(κh)

(8.32)

where =

ǫζ µ

(8.33)

The dimensionless electroosmotic velocity profile in the microchannel is a strong function of the dimensionless channel gap as represented by κh. For κh < 5, the dimensionless velocity shows strong dependence on the transverse position, y/ h. However, for large κh values, the velocity profile becomes fairly flat in the central region of the flow channel. In practice, for channels of h > 10 microns with a (1:1) electrolyte concentration as low as 10−6 M, the value of κh is quite large and the velocity profile is fairly flat. In the case of h < 1 µm coupled with free charge carrying organic liquids, one would expect to achieve low values of κh. 8.2.3 Volumetric Flow Rate The volumetric flow per unit width of the slit microchannel is given by Q=2



0

h

ux (y) dy

(8.34)

8.2 ELECTROOSMOTIC FLOW IN A SLIT CHARGED MICROCHANNEL

239

The units of Q are m3 /(ms). Substituting for ux (y) from Eq. (8.30) in Eq. (8.34) and performing the integration lead to   2hǫEx ζ tanh(κh) 2h3 px − Q= 1− 3µ µ κh

(8.35)

Denoting the channel cross-sectional area per unit width by Ac , where Ac = 2h Equation (8.35) becomes   Q h2 p x ǫEx ζ tanh(κh) = − 1− Ac 3µ µ κh

(8.36)

The term Q/Ac represents the volumetric fluid flow rate per unit width under pressure and electric potential gradients. The first term of Eq. (8.36) is the volumetric flow rate per unit width, Qpress /Ac , due to a pressure gradient. For a given gap and liquid viscosity, Qpress /Ac is directly proportional to the pressure gradient. Such a relationship is to be expected for a flow process governed by a linear differential equation. For given px /µ, the pressure driven volumetric flow rate per unit width is much influenced by the channel gap spacing, 2h. The second term of Eq. (8.36) gives the volumetric flow rate per unit width, Qel /Ac , for the case of electroosmotic flow that is driven by the electric potential gradient. The term Qel /Ac is directly proportional to ǫζ Ex /µ, which can be considered as the effective electric driving force. Once again such a linear relationship is to be expected. The dimensionless electroosmotic flow is given by tanh(κh) Qel =1− (−Ac Ex ) κh

(8.37)

For large values of κh, tanh(κh) → 1 and Eq. (8.37) indicates that the dimensionless electroosmotic volumetric flow rate approaches unity at large values of κh. For small values of κh < 1, Eq. (8.37) gives (κh)2 Qel = (−Ac Ex ) 3

(8.38)

indicating that there is minimal electroosmotic flow for small values of κh. Figure 8.5 depicts the scaled electroosmotic flow contributions obtained using Eqs. (8.37) and (8.38). It is clear from Figure 8.5 that the greatest advantage in having electroosmotic flow is for the case of large κh (i.e., a thin electric double layer).

240

FLOW IN MICROCHANNELS

Figure 8.5. Variation of the electroosmotic volumetric flow rate per unit channel width with the electric double layer thickness κh. The dashed line represents the approximate expression valid for small values of κh.

8.3 ELECTROOSMOTIC FLOW IN A CLOSED SLIT MICROCHANNEL Electrophoretic measurements are normally conducted in rectangular channels having a large aspect ratio. In such electrophoretic measurements, the movement of the charged particles are monitored in order to evaluate their surface charge. An electrophoretic cell is shown in Figure 8.6. The electrodes are on the extreme ends of the

Figure 8.6. An electrophoretic cell.

8.3 ELECTROOSMOTIC FLOW IN A CLOSED SLIT MICROCHANNEL

241

Figure 8.7. Schematic of a velocity profile in streaming potential flow in a closed slit microchannel having a negative zeta potential.

cell and the cell itself is sealed. In other words, there is no net flow within a given cross-section of the cell. As a rectangular cell with a large aspect ratio can be approximated by a one-dimensional flow problem, that of a gap formed by two parallel plates, the analysis presented here can be used to study flows in an electrophoretic cell. For a closed narrow gap channel, as shown in Figure 8.7, the volumetric flow per unit width is zero. Setting Q to zero, Eq. (8.36) yields the pressure gradient that has to be developed by the liquid to accommodate the zero net flow,   tanh(κh) 3ǫEx ζ 1− px = h2 κh

(8.39)

The pressure gradient has a maximum absolute value when κh → ∞. Substituting for px in Eq. (8.30) yields the local velocity in the microchannel for the special case of Q = 0,      y 2   ux tanh(κh) 3 cosh(κy) 1− =− 1− + 1− (− Ex ) 2 h κh cosh(κh)

(8.40)

The term ux /(− Ex ) represents a convenient dimensionless local axial velocity. Figure 8.8 shows the dimensionless local velocity in a microchannel for different values of κh. The overall velocity profile is a superposition of the electroosmotic flow and the counteracting pressure driven flow. For a negative value of the surface potential, i.e., ζ < 0, the flow near the wall is along the positive axial direction. Although the total flow rate is zero, the local velocities at different normal distances from the channel wall evolve from variations of the electroosmotic velocity along the positive axial direction and the counteracting pressure driven velocity along the negative axial direction. At the microchannel wall, both electroosmotic and pressure driven velocities are zero (no-slip). Within the electric double layer region adjacent to the wall, the electroosmotic velocity increases more rapidly than the pressure driven velocity as one moves away from the wall. Thus, the net velocity in the

242

FLOW IN MICROCHANNELS

Figure 8.8. Variation of the dimensionless local velocity with transverse position in a slit microchannel during electroosmotic flow for different values of the scaled half-channel height κh.

positive axial direction increases as one moves away from the the wall. Beyond the electrical double layer, the electroosmotic velocity becomes virtually constant, although the pressure driven velocity profile continues to increase in a parabolic manner up to the axis of symmetry of the microchannel. This causes a reversal in the velocity direction, and near the central core of the microchannel, the net velocity is along the negative axial direction. Thus, near the microchannel wall, the electroosmotic flow dominates, whereas near the channel centerline, the pressure driven flow dominates. The very fact that Q is set to zero, i.e., no net flow, implies that there must exist a flow reversal in the channel gap in order to satisfy the mass conservation and, consequently, a location at which the local velocity is zero. The location of zero local velocity is referred to as the stationary surface and it is on that surface that electrophoretic measurements are made. If one is to place a charged particle at the stationary surface, then the electrophoretic velocity of the particle is not influenced by the electroosmotic velocity of the electrolyte solution. Consequently, electrophoresis devices are designed to observe particle motion along such a surface for the measurement of the electrophoretic particle velocity with the subsequent evaluation of the particle surface charge (Shaw, 1980). The location of the stationary surface is easily obtained by setting ux to zero in Eq. (8.40). For the special case of κh → ∞, one obtains y 1 = √ = 0.577 h 3

(8.41)

Figure 8.9 shows the location of the stationary plane as √ a function of κh. It can be noted that for κh > 100, the stationary plane is nearly 1/ 3.

8.4 EFFECTIVENESS OF ELECTROOSMOTIC FLOW

243

Figure 8.9. Location of the stationary plane in a slit microchannel during electroosmotic flow.

8.4 EFFECTIVENESS OF ELECTROOSMOTIC FLOW The effectiveness of electroosmotic flow as compared to pressure driven flows can be estimated using the volumetric flow ratio, VFR, defined as VFR =

volumetric flow rate due to electroosmosis volumetric flow rate due to pressure

(8.42)

Making use of Eq. (8.36) one can write VFR = −

  tanh(κh) 3ǫEx ζ 1 − h2 p x κh

(8.43)

For given values of ǫ, Ex , ζ , and px , we obtain   tanh(κh) 1 VFR ∝ 2 1 − h κh

(8.44)

For the case of κh ≫ 1, i.e., thin double layers, the term tanh(κh)/(κh) becomes zero and one obtains 1 VFR ∝ 2 h The above expression clearly shows that electroosmotic flow is most effective for very narrow channels with κh ≫ 1. Consequently, in practice for the case of κh ≫ 1, one uses an applied electric potential to affect flow in a narrow channel rather than using an imposed pressure gradient. For the case of κh ≪ 1, where the microchannel is characterized by overlapping double layers, one can expand the term tanh(κh) in Eq. (8.43) in a Taylor series,

244

FLOW IN MICROCHANNELS

yielding   3ǫEx ζ (κh) − (κh)3 /3 + O[(κh)5 ] ǫEx ζ 2 VFR = 2 1− κ ≃ h px κh px

(8.45)

This indicates that in the limit κh ≪ 1, the VFR becomes independent of the channel height and assumes a large constant value depending on the double layer parameter κ.

8.5 ELECTRIC CURRENT IN ELECTROOSMOTIC FLOW IN SLIT CHANNELS In the previous sections, we have developed expressions for the electric potential and the flow velocity distributions for the case of flow in slit microchannels. The major simplification in the analysis was the assumption of low surface potentials where the Debye–Hückel approximation was employed. During flow in a microchannel, electric current can flow within the channel itself relative to the external circuit that provided the external potential. In this case, both the potential gradient and the electric current are non-zero quantities. Such a situation occurs in purely electroosmotic flow. For purely pressure driven flows, where the two ends of a channel are not connected by an external circuit, there will be no current flow but there will be an induced electric potential gradient. Such a flow situation is referred to as streaming potential flow. In this section, we will derive an expression for the total electric current that will be used to inter-relate the various types of flows with each other. The current density vector, in A/m2 , is obtained from Eq. (6.69) as i = eu


k

z k nk − e


k

Dk zk ∇nk −

e2 ∇φ
2 z D k nk kB T k k

(8.46)

where φ represents the total electric potential. The flow in the slit microchannel is unidirectional where the fluid and current flows are in the x-direction only. Consequently, we need only to consider the current in the x-direction. Equation (8.46) in x− direction is given by ix = eux


k

z k nk − e


k

Dk zk

∂nk e2 ∂φ
2 − z Dk nk ∂x kB T ∂x k k

(8.47)

Equation (8.47) expresses the current density (A/m2 ) at a given location in the channel. As discussed earlier in Chapter 6, the first term of Eq. (8.47) is the current density due to the bulk motion of the fluid, the second term is due to ionic diffusion, and the third term is due to migration as affected by the potential gradient. In the present flow system, it is assumed that there is no variation in the ionic concentration along the length of the channel. Consequently, ∂nk /∂x = 0, and the second term of Eq. (8.47) drops out.

8.5 ELECTRIC CURRENT IN ELECTROOSMOTIC FLOW IN SLIT CHANNELS

245

The total current flow per unit width of the slit channel (A/m) is given by

Ix = I = 2

h

0

ix dy

(8.48)

Making use of Eqs. (8.1) and (8.47), the current per unit width of the slit channel becomes

h
2e2 Ex h
2 zk nk dy + ux zk Dk nk dy (8.49) I = 2e kB T 0 0 where the local fluid velocity is given by Eq. (8.30) can be written as   y 2  ǫE h2 p x x − [ζ − ψ] 1− ux (y) = 2µ h µ

(8.50)

The first term in Eq. (8.49) is the contribution to the current due to convective transport  and the presence of the charged channel surface where electroneutrality term zk nk is non-zero. The convection current can be thought of as being due to the flow of the excess ions in the mobile double layer region close to the surface caused by the pressure driven liquid flow through the channel. For the present, we will refer to it as the convection transport current, It . It is also referred to as the streaming current. The second term of Eq. (8.49) is due to the electric conduction within the liquid along the channel. It is referred to in the literature as the conduction current, and we will denote it by Ic . These two contributions, It and Ic , to the total current, I , play a major role when dealing with pressure induced flows as is the case for streaming potential flow. In this analysis we assume that there is no current due to the channel wall material itself or due to a current within the Stern layer, i.e., in the layer between the channel wall and the shear plane. Note that the boundary conditions of the electrokinetic problem are applied at the shear plane in the present analysis. In other words, we are completely ignoring any electrical effects arising in the region between the channel walls and the shear plane. Let us first consider the case of conduction current, Ic , where 2e2 Ex Ic = kB T



h 0




zk2 Dk nk dy

(8.51)

Let us consider the special case of a symmetric (z : z) electrolyte where the ionic species have the same diffusion coefficients, where D+ = D− = D. For a (z : z) 2 2 = z2 and Eq. (8.51) becomes = z− electrolyte, one has z+ Ic =

2e2 z2 DEx kB T



0

h

[n+ + n− ] dy

(8.52)

246

FLOW IN MICROCHANNELS

Making use of the Boltzmann distribution, Eqs. (8.7) and (8.8), Eq. (8.52) becomes 4e2 z2 Dn∞ Ex Ic = kB T



h 0



zeψ cosh kB T



(8.53)

dy

One can substitute the electric potential distribution, ψ(y), obtained from a solution of the Poisson–Boltzmann equation in Eq. (8.53) to obtain the conduction current. To keep the approach analytical, we will substitute for ψ using Eq. (8.17), which was obtained using the Debye–Hückel approximation, valid for low potentials. In doing so, however, we first note that the integrand on the right hand side of Eq. (8.53), namely cosh(zeψ/kB T ), should also be expressed as a Taylor series, retaining only the leading order terms in the resulting expansion. Consequently, Eq. (8.53) simplifies to 4e2 z2 Dn∞ Ex Ic = kB T





1 1+ 2

h 0



zeψ kB T

2

+O



zeψ kB T

4 

dy

(8.54)

Now, substituting Eq. (8.17) in Eq. (8.54), the expression for the conduction current becomes 4e2 z2 Dn∞ Ex Ic = kB T



h 0



1 1+ 2



zeζ kB T

2

cosh2 (κy) cosh2 (κh)



dy

(8.55)

dY

(8.56)

Letting Y = y/ h, we can write for the conduction current 4e2 z2 Dn∞ hEx Ic = kB T



0

1



1 1+ 2



zeζ kB T

2

cosh2 (κh · Y ) cosh2 (κh)



We now note that for a (z : z) electroneutral electrolyte solution where ni = ni∞ and Di = D, the electric conductivity is given by Eq. (6.74) as σ∞ =

2e2 z2 Dn∞ kB T

with units of S/m or A/Vm. Here we use the symbol σ ∞ instead of σ to emphasize that the conductivity refers to the bulk electroneutral electrolyte. Using the above equation for the electric conductivity, one can write Eq. (8.56) as Ic = 2σ ∞ hEx Fcs

(8.57)

8.5 ELECTRIC CURRENT IN ELECTROOSMOTIC FLOW IN SLIT CHANNELS

247

with    1 zeζ 2 cosh2 (κh · Y ) Fcs = 1+ dY 2 kB T cosh2 (κh) 0     1 1 zeζ 2 tanh(κh) + =1+ 4 kB T κh cosh2 (κh)

1



(8.58)

where Fcs is a factor to account for the non-electroneutrality of the electrolyte solution due to the presence of the charged channel surfaces. Figure 8.10 shows the values of Fcs for different dimensionless surface potentials, (zeζ /kB T ) as a function of the scaled channel height, κh, where the scaling is done with respect to the Debye length, κ −1 . As the solution for the electric potential was obtained for zeζ /kB T < 1 (low surface potentials), by keeping the restriction of zeζ /kB T < 1, the value of Fcs can be set to unity for κh > 5. However, if the analytical solution as obtained for zeζ /kB T < 1 is extended for higher values of zeζ /kB T , then one cannot set Fcs to unity except for large values of κh, say κh > 50. Now we can turn our attention to the first term of Eq. (8.49), and deal with the convection transport current. The convection transport current, It , is given by It = 2e



0

h

ux




zk nk dy

(8.59)

Recognizing that e




zk nk = ρf

(8.60)

Figure 8.10. Variation of the function Fcs given in Eq. (8.58) with κh for different scaled surface potentials.

248

FLOW IN MICROCHANNELS

and using the Poisson equation, Eq. (8.4), the convection transport current becomes It = −2ǫ

h



ux

0

d 2ψ dy dy 2

(8.61)

A straightforward approach would be to substitute the expressions for ux and ψ in the integral of Eq. (8.61) and to perform the integration. A different approach, based on integration by parts, may be followed to simplify the problem considerably (Hunter, 1981; Erickson et al., 2000). Integration by parts of Eq. (8.61) leads to

It = −2ǫ



dψ ux dy

y=h y=0

−



h 0

dψ dux dy



(8.62)

The first term of Eq. (8.62) is zero since dψ =0 dy

at y = 0

and

ux = 0

at y = h

Equation (8.62) can be written as It = 2ǫ



h

0

dux dψ dy

(8.63)

Differentiation of the velocity field given by Eq. (8.50) and its substitution in Eq. (8.63) yield 

 dψ 2ǫ h dψ It = −px y + ǫEx dy · µ 0 dy dy leading to 2ǫ It = µ



h

0



dψ + ǫEx −px y dy



dψ dy

2 

dy

(8.64)

The integrals of Eq. (8.64) are given by

0

h

  dψ tanh(κh) y dy = hζ 1 − dy κh

(8.65)

and

h 0



dψ dy

2

  κζ 2 κh dy = tanh(κh) − 2 cosh2 (κh)

(8.66)

249

8.5 ELECTRIC CURRENT IN ELECTROOSMOTIC FLOW IN SLIT CHANNELS

The convection transport current is then given by     2ǫhpx ζ 1 tanh(κh) tanh(κh) It = − − (8.67) 1− + µκ 2 h 2 Ex µ κh κh cosh2 (κh) The total current can now be expressed as     2ǫhpx ζ 1 tanh(κh) tanh(κh) I =− − 1− + µκ 2 h 2 Ex µ κh κh cosh2 (κh) + 2σ ∞ hEx Fcs

(8.68)

For convenience, let α1 = 1 − and α2 =

tanh(κh) κh

(8.69-a)

1 tanh(κh) − κh cosh2 (κh)

(8.69-b)

Using the above, the total current expression becomes I =−

2ǫhpx ζ α1 + µκ 2 h 2 Ex α2 + 2hσ ∞ Ex Fcs µ

(8.70)

In the limit of κh ≫ 1, Fcs and α1 approach unity and α2 approaches zero. Noting that the current is caused by two driving forces, namely, the electric field, Ex , and the pressure gradient, px , Eq. (8.70) can be written in the terminology of non-equilibrium thermodynamics as (8.71)

I = L11 Ex + L12 px where L11 = 2hσ and L12 = −

∞



µκ 2 2 α2 Fcs + 2σ ∞



  2ǫhζ tanh(κh) 2ǫhζ α1 = − 1− µ µ κh

(8.72)

(8.73)

Let us now recast the expression for Q, Eq. (8.35), as Q=

2h3 px 2ǫhEx ζ − α1 3µ µ

(8.74)

Once again, noting that the volumetric flow is caused by the two driving forces Ex and px , we can write Eq. (8.74) in the form of a non equilibrium thermodynamic relationship as Q = L21 Ex + L22 px

(8.75)

250

FLOW IN MICROCHANNELS

where L21

  2ǫhζ tanh(κh) 2ǫhζ α1 = − =− 1− µ µ κh

(8.76)

and L22 =

2h3 3µ

(8.77)

Comparing Eqs. (8.73) and (8.76) we immediately note that the coefficients L12 and L21 are identical. This validates the Onsager reciprocal relationship between the current and the volumetric flow rate. We will revisit these Onsager relationships in the next section. For the case of a flow purely driven by an electric potential gradient, Ex , in the absence of a pressure gradient, i.e., px = 0, Eqs. (8.71) and (8.75) provide 

Q I



= px =0

L12 ǫζ L21 [1 − tanh(κh)/(κh)]  = =− ∞ L11 L11 µσ Fcs + µκ 2 2 α2 /(2σ ∞ )

(8.78)

where the final expression was obtained by using Eqs. (8.73) and (8.72) for L12 and L11 , respectively. The term (Q/I )px =0 is often referred to as the electroosmotic coefficient. After substituting the expressions for Fcs and α2 in Eq. (8.78), and expanding the resulting expression as a Taylor series in zeζ /kB T , one can obtain a simplified relationship for the electroosmotic coefficient. The resulting expression, neglecting terms of the order of O[(zeζ /kB T )3 ] is 

Q I



px =0

    tanh(κh) ǫζ zeζ 3 ≃− ∞ 1− +O µσ (κh) kB T

(8.79)

Equation (8.79) relates the liquid volumetric flow to the current flow for a purely electroosmotic flow, a situation depicted in Figure 8.11.

Figure 8.11. Purely electroosmotic flow in a slit microchannel. There is no pressure gradient across the channel. The flow is caused solely by the electric field gradient set up between the electrodes at the channel entrance and exit.

8.6 STREAMING POTENTIAL IN SLIT CHANNELS

251

8.6 STREAMING POTENTIAL IN SLIT CHANNELS When a liquid is forced through a channel, the ionic charges in the mobile part of the electric double layer near the surface of the channel walls are convected in the flow direction toward the channel exit. Such a current constitutes the streaming current. However, as there is no external electric connection between the channel inlet and the exit, the accumulation of charge sets up an electric field such that a current flows in the opposite direction through the bulk of the liquid. This current is called the conduction current. At steady state, the net current is zero whereby the streaming and conduction currents sum up to zero. The electric potential that is induced by the flow is called the streaming potential and the pressure induced flow is referred to as streaming potential flow. To analyze the streaming potential flow, we simply set the current to zero. Setting I = 0 in Eq. (8.71) provides,   Ex L12 =− (8.80) px I =0 L11 Comparing Eqs. (8.78) and (8.80) shows that     Ex Q =− px I =0 I px =0

(8.81)

This is a fundamental relationship originally shown by Mazur and Overbeek (1951) to be a consequence of the Onsager principle of reciprocity for irreversible phenomena. Substituting the expressions for L11 and L12 in Eq. (8.81) provides     Q [1 − tanh(κh)/(κh)] ǫζ Ex =− (8.82) = px I =0 I px =0 µσ ∞ [Fcs + µκ 2 2 α2 /(2σ ∞ )] Once again, to the leading order in zeζ /kB T , the above expression simplifies to         Ex Q tanh(κh) ǫζ zeζ 3 (8.83) =− 1− +O ≃ px I =0 I px =0 µσ ∞ (κh) kB T In the limit of κh ≫ 1, we can write     Ex Q ǫζ =− = px I =0 I px =0 µσ ∞

(8.84)

Due to the nature of the functions Fcs and α2 , although Eq. (8.83) is strictly valid for small values of zeζ /kB T < 1, Eq. (8.84) is valid for all values of zeζ /kB T as long as κh ≫ 1. Note that the streaming potential is manifested as an electric potential difference, V = Vinlet − Voutlet , (Volts) over a certain length, L, of the microchannel. Using Ex =

V L

252

FLOW IN MICROCHANNELS

one can obtain from Eq. (8.83) V =

  ǫζ L tanh(κh) 1 − px µσ ∞ (κh)

(8.85)

Note that the potential difference is measured under conditions that the total current, I , is zero.

8.7 ELECTROVISCOUS FLOW IN SLIT MICROCHANNELS When a liquid is forced through a microchannel under an applied pressure gradient (in the absence of an externally applied electric field), an induced streaming electric field, Ex , is established and the volumetric flow rate is given by Eq. (8.74). The current flow is zero, as there is no external electric connections between the inlet and outlet of the slit microchannel. The streaming electric field is now given by Eq. (8.82) as Ex is no longer an independent variable imposed by the experimentalist. The field is related to the pressure gradient, px , as Ex =

ǫζ [1 − tanh(κh)/(κh)] px ∞ µσ [Fcs + µκ 2 2 α2 /(2σ ∞ )]

for I = 0

(8.86)

Substituting the above expression for Ex in Eq. (8.74), the volumetric flow rate per unit width is given by   2h3 px α12 3ǫ 2 κ 2 ζ 2 Q= 1− 3µ µσ ∞ (κh)2 [Fcs + µκ 2 2 α2 /(2σ ∞ )]

(8.87)

Once again, to a leading order term in zeζ /kB T , one can simplify Eq. (8.87) to obtain Q=

  2h3 px 3ǫ 2 ζ 2 κ 2 2 α 1− 3µ µσ ∞ (κh)2 1

(8.88)

The first term of Eq. (8.88), including the multiplier (2h3 px /3µ) is due to the imposed pressure gradient. It is a positive quantity. The second term on the right hand side of the above expression is due to the induced potential (streaming potential). This term is positive irrespective of the sign of the surface potential ζ . Consequently, the induced potential gives rise to a reduced volumetric flow rate for a given applied pressure, irrespective of its sign. This reduction in volumetric flow rate gives the appearance of an increased liquid viscosity, hence, the term “electroviscous effects”. The reduced rate of flow results in an apparent viscosity µa defined by Q=

2h3 px 3µa

(8.89)

8.8 ELECTROOSMOTIC FLOW IN A CIRCULAR CHARGED CAPILLARY

253

Figure 8.12. Variation of the apparent viscosity, µa /µ, with κh for different values of the parameter β. The results are depicted for a scaled surface potential of s = zeζ /kB T = 1.0 on the walls of the slit microchannel.

Equating for Q in Eqs. (8.88) and (8.89) leads to  −1   µa 3ǫ 2 ζ 2 κ 2 2 3ǫ 2 ζ 2 κ 2 2 ≃ 1 + = 1− α α µ µσ ∞ (κh)2 1 µσ ∞ (κh)2 1

(8.90)

where the final term was obtained using a Taylor expansion. Figure 8.12 shows the variation of µa /µ with κh for different values of the parameter β, where β=

2 µκ 2 ǫ2ζ 2κ 2 = ∞ µσ σ∞

(8.91)

The increase in µa /µ is prominent at small and intermediate values of κh.

8.8 ELECTROOSMOTIC FLOW IN A CIRCULAR CHARGED CAPILLARY The analysis for slit microchannels described so far in this chapter delineates the fundamental principles of electroosmotic flow in narrow microchannels. However, for realistic scenarios involving electrokinetic transport in microchannels, a more relevant modelling approach would be to employ circular cylindrical geometries to model capillary flows. For example, an electrolyte solution in a porous medium can be made to move when an electric field is applied. In order to understand electroosmotic flow in such a porous medium or in capillary tubes of varying cross-sections, it is

254

FLOW IN MICROCHANNELS

Figure 8.13. Geometry of a circular cylindrical microchannel used for modelling electroosmotic flow.

first necessary to be able to analyze the simple situation of electroosmotic flow in a straight circular capillary tube or a circular cylindrical microchannel. In this analysis, we shall present electroosmotic flow in a straight tube having a low surface potential. The flow to be considered is fully developed; i.e., no end effects are present. By and large, the analysis is that given by Rice and Whitehead (1965). Improvements to include a high surface potential are given by Levine et al. (1975). Furthermore, since the details of the theoretical principles are quite similar to the slit microchannel analysis, except that the present analysis is based on a cylindrical coordinate system, we will present a condensed version of the pertinent derivations. Consider a circular cylindrical microchannel of radius a with a negatively charged surface bearing a surface potential of ζ ,1 as illustrated in Figure 8.13. The coordinate system used is cylindrical with r representing the radial direction and x representing the axial direction. Consider a symmetric (z : z) electrolyte flowing in the capillary having a surface potential ζ . Let the total potential at a point (r, x) be φ = φ(r, x) = ψ(r) + (φo − xEx )

(8.92)

where ψ(r) is the potential due to the double layer at the equilibrium state corresponding to no fluid motion and no applied external field, φo is the value of the imposed potential at x = 0, and φo − xEx is the potential at any axial location in the capillary due to the applied electric field Ex in the absence of the double layer. Equation (8.92) is identical to Eq. (8.2) except that it is written for a cylindrical coordinate system. The Poisson equation defining the potential distribution in cylindrical coordinates is given by ρf ∇ 2φ = − ǫ 1

The surface potential ζ represents the potential at the shear plane, as in the case of a slit microchannel.

8.8 ELECTROOSMOTIC FLOW IN A CIRCULAR CHARGED CAPILLARY

255

or 1 ∂ r ∂r



r

∂φ ∂r



ρf ∂ 2φ =− ∂x 2 ǫ

+

(8.93)

When Eq. (8.92) is substituted, one obtains 1 d r dr



r

dψ dr



=−

ρf ǫ

(8.94)

where the free charge density, ρf , is a function of r, and it is given by ρf =




ezk nk

(8.95)

k

For a symmetric (z : z) electrolyte, invoking the Boltzmann distribution, Eqs. (8.7) and (8.8), for the ion concentrations, one obtains the Poisson–Boltzmann equation, written as     1 d dψ zeψ r = κ 2 sinh (8.96) r dr dr kB T where κ is the inverse Debye screening thickness given by Eq. (8.14). If zeψ/kB T is small (i.e., ψ ≤ 25mV), we can write sinh(zeψ/kB T ) ≈ zeψ/kB T , whereby the Debye–Hückel approximation is invoked and one obtains the linearized Poisson–Boltzmann equation: 1 d r dr



r

dψ dr



= κ 2ψ

(8.97)

The above Poisson–Boltzmann equation can be solved subject to the boundary conditions dψ = 0 (axisymmetry) (8.98) at r = 0, dr and at r = a, ψ = ζ

(8.99)

where a is the radius of the capillary tube. The solution to Eq. (8.97) satisfying condition (8.98) is ψ = BIo (κr)

(8.100)

where Io is the zeroth-order modified Bessel function of the first kind and B is a constant of integration. The second boundary condition, namely, ψ = ζ at r = a, leads to Io (κr) ψ =ζ (8.101) Io (κa)

256

FLOW IN MICROCHANNELS

With the above solution, one has for the free charge density distribution ρf (r) = −ǫκ 2 ψ = −ǫκ 2 ζ

Io (κr) Io (κa)

(8.102)

Up to this point we have solved for the electric parameters using the Boltzmann distribution for the ionic concentration along the radial direction. Having obtained the expressions for the electrostatic potential in the radial direction and for the charge distribution, we are prepared to solve the flow equation for the velocity profile. The modified Navier–Stokes equation in the axial flow direction is given by   1 d ∂φ dux dp µ + ρf (8.103) r = r dr dr dx ∂x with Ex = −

∂φ ∂x

(8.104)

Using Eq. (8.102), the momentum equation becomes (setting px = −dp/dx) 1 d µ r dr

  Io (κr) dux r = −px + ǫκ 2 ζ Ex dr Io (κa)

(8.105)

with the boundary conditions ux = 0

at r = a

(8.106)

and dux =0 dr

at r = 0

(8.107)

The solution of Eq. (8.105), subject to Eqs. (8.106) and (8.107), is given by    r 2  ǫζ  a 2 px Io (κr) ux (r) ≡ ux = − Ex 1− 1− 4µ a µ Io (κa)

(8.108)

The above result shows that the velocity of the electrolyte solution is the sum of the Poiseuille flow term and the electrokinetic term. Equations (8.101) and (8.108) define the solution to the electroosmotic flow in the circular capillary. We can now make use of the general solution given by Eq. (8.108) and derive interesting limiting cases. Consider the case of px = 0, i.e., there is no imposed pressure gradient as is the case for a capillary connecting two large reservoirs as shown in Figure 8.14. Equation (8.108) then gives for px = 0 ux,el

  Io (κr) ǫζ 1− Ex =− µ Io (κa)

(8.109)

8.8 ELECTROOSMOTIC FLOW IN A CIRCULAR CHARGED CAPILLARY

257

Figure 8.14. A circular cylindrical microchannel connecting two reservoirs having the same hydrostatic head (same liquid depth). In this case, one obtains a purely electroosmotic flow in the capillary microchannel when an electric field is applied across the electrodes.

For given κa and ǫ, ux,el increases linearly with the surface potential ζ and the electric field strength Ex . Also, ux,el is inversely proportional to the fluid viscosity (other parameters being constant). Let Io (κr) =A (8.110) Io (κa) The term in the square brackets of Eq. (8.109) is always positive. For a negative surface potential the axial velocity ux,el becomes positive. 8.8.1 Thin Double Layers: Helmholtz–Smoluchowski Equation, κa ≫ 1 In the absence of a pressure gradient, i.e., px = 0, and for thin double layers, κa ≫ 1, the term Io (κr)/Io (κa) = 0, in which case Eq. (8.109) becomes ux,el = −

ǫζ Ex = constant
= f (r) µ

(8.111)

a result known as the Helmholtz–Smoluchowski equation – a classical equation for the flow of an electrolyte past a charged surface under the influence of an electric field along the surface. Here the fluid moves as a plug, as if the fluid slips at the wall. A plot of Eq. (8.109) is shown in Figure 8.15 for various values of κa. To keep the nomenclature consistent with the slit microchannel analysis presented earlier, we will define ǫζ = µ

8.8.2 Thick Double Layers, κa ≪ 1 For the special case of κa ≪ 1, i.e., overlapping electric double layers with px = 0, one can write a series expansion for Io in terms of κa and κr to obtain ux,el

  r 2  2 = − Ex (κa) 1 − 4 a

(8.112)

258

FLOW IN MICROCHANNELS

Figure 8.15. Dimensionless electroosmotic velocity profiles for px = 0 for a circular capillary, where = ǫζ /µ.

Note that if κa → 0 then ux,el → 0 and we have no electroosmotic flow for substantially overlapping double layers. Equation (8.112) gives a parabolic velocity profile. The term − Ex (κa)2 /4 is equivalent to (a 2 px /4µ), which characterizes the pressure driven Poiseuille flow. The Poiseuille type flow characteristics at low values of κa are due to the large overlap of the double layer with a virtually constant net free charge density across the capillary. This situation gives rise to a liquid body force similar to that of a pressure gradient. The volumetric flow rate is given by Q = 2π



a

0

rux dr

(8.113)

For the special case of px = 0, we can write   2I1 (κa) Qel = − Ac Ex 1 − (κa)I0 (κa)

(8.114)

where I1 is first-order modified Bessel function of the first kind, Ac is the crosssectional area of the capillary tube equal to π a 2 , and Qel is the volumetric flow rate due to the electroosmotic flow. A dimensionless form of Qel is shown in Figure 8.16. As can be observed from Figure 8.16, the greatest advantage in having electroosmotic flow is for the case of large κa (i.e., a thin double layer).

8.8 ELECTROOSMOTIC FLOW IN A CIRCULAR CHARGED CAPILLARY

259

Figure 8.16. Variation of the scaled electroosmotic volumetric flow rate for a circular capillary with κa for px = 0.

For the general case (px
= 0), the total volumetric flow rate Q is given by   px a 2 2I1 (κa) Q= Ac − Ac Ex 1 − (8.115) 8µ (κa)I0 (κa) When Q = 0, i.e., there is no net flow, the ratio of the developed pressure gradient to the electric field is given by    px 8ǫζ 2I1 (κa) = 1− (8.116) Ex a2 (κa)I0 (κa) The pressure gradient px becomes the electroosmotic pressure. For κa ≫ 1 (i.e., a thin double layer), Eq. (8.116) becomes px 8ǫζ = 2 Ex a

(8.117)

For the special case of Q = 0, one can make use of Eq. (8.116) to eliminate px and evaluate the local axial velocity profile as given by Eq. (8.108) to obtain       r 2   ux 2I1 (κa) I0 (κr) 1− = 2 1− − 1− (8.118) Ex a (κa)I0 (κa) I0 (κa) For the special case of a thin double layer, i.e., κa ≫ 1, Eq. (8.118) gives   r 2  ux = 1−2 Ex a Equation (8.119) is valid for (r/a) < 1.

(8.119)

260

FLOW IN MICROCHANNELS

Figure 8.17. Variation of the dimensionless axial velocity with radial position for the special case of zero volumetric flow rate at various values of κa for a circular capillary.

The dimensionless velocity profiles, −

µux ux =− ǫζ Ex Ex

given by Eq. (8.118) are shown in Figure 8.17. As the volumetric flow rate Q is zero, the axial velocity ux has to change direction in order to conserve mass balance. As described in the case of a slit microchannel, the overall velocity profile evolves from the competition between the electroosmotic flow and the counteracting pressure driven flow. The electroosmotic flow has a dominant effect near the microcapillary wall, inside the electric double layer, while the pressure driven flow dominates in the core region. An important feature of Figure 8.17 is the presence of a cylindrical shell within the capillary tube where the electrolyte solution velocity is zero. The radial position of this shell is usually referred to as the stationary surface, which is similar to the stationary plane obtained for the case of a slit microchannel. If one places a charged particle at the stationary surface, then the electrophoretic velocity of the particle is not influenced by the velocity of the electrolyte solution. Consequently, electrophoresis devices are designed to observe particle movements along such a surface for the measurement of electrophoretic particle velocity and the subsequent evaluation of the particle surface charge (Shaw, 1980). Table 8.1 gives the values of the stationary radial position, Rst (made dimensionless with √ a). In the limit of κa → ∞, the dimensionless radial position becomes Rst = 1/2. For a capillary radius of about 1 mm, κa can be considered to be very large for most electrolyte solutions and Rst = 0.7071.

8.8 ELECTROOSMOTIC FLOW IN A CIRCULAR CHARGED CAPILLARY

261

TABLE 8.1. Dimensionless Radial Position, Rst , of the Stationary Surface for a Low Surface Potential Capillary. κa

Rst

1 2 5 10 20 50 100 200 500 ∞

0.57867 0.58241 0.60216 0.63500 0.66851 0.69238 0.69989 0.70354 0.70569 0.70711

For the general case of electroosmotic flow, the effectiveness of the electric field can be estimated using the volumetric flow ratio, VFR, defined as VFR =

flow rate due to electroosmosis flow rate due to pressure

(8.120)

From Eq. (8.115) one can write VFR = −

  8ǫζ Ex 2I1 (κa) 1 − a 2 px κaI0 (κa)

For given values of ζ , ǫ, Ex and px , Eq. (8.121) shows that   2I1 (κa) 1 VFR ∝ 2 1 − a κaI0 (κa)

(8.121)

(8.122)

For the case of κa ≫ 1, i.e., thin double layers, the numerator on the right side of Eq. (8.122) becomes unity and 1 (8.123) VFR ∝ 2 a In this case of thin double layers, the electric field becomes very effective in driving the flow. For the case of κa ≪ 1 where the capillary tube is characterized by overlapping double layers, the VFR attains a large value independent of the capillary radius. This can be demonstrated by expanding the Bessel functions of Eq. (8.121) in a Taylor series of κa, and simplifying the resulting expression. The procedure yields VFR ≃ −

ǫEx ζ κ 2 px

The same limiting cases were encountered for the slit microchannel.

(8.124)

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FLOW IN MICROCHANNELS

8.8.3

Current Flow in Electroosmosis

The total current due to the electroosmotic flow is given by I = 2π



a

(8.125)

ix r dr

0

where ix is the local current density (A/m2 ). Making use of Eq. (6.69) one can write for the total current in x-direction

I = 2π

0

−

a

ux

a


 ∂nk r dr Dk zk ∂x k

ezk nk r dr − 2π e



∂φ
 2 zk Dk nk r dr ∂x k

k

e2 2π kB T






a

0

0

(8.126)

For a fully developed flow, there is no axial variation in the ionic concentration, and hence, the second term of Eq. (8.126) becomes zero. Recognizing that the charge density is given by
ρf = e zk nk k

and setting ∂φ = −Ex ∂x Equation (8.126) becomes I = 2π



0

a

2π e2 ux ρf r dr + Ex kB T



a

0








zk2 Dk nk r dr

k

(8.127)

2 2 Let Dk = D, i.e., D+ = D− = D. For a z : z electrolyte, one has z+ = z− = z2 . Using these simplifications in Eq. (8.127) yields

I = 2π



0

a

ux ρf r dr +

2π e2 z2 DEx kB T



a

nk r dr

(8.128)

0






zeψ kB T

2 

(8.129)

k

Note that
k



zeψ nk = 2n∞ cosh kB T



≃ 2n∞



1 1+ 2

8.8 ELECTROOSMOTIC FLOW IN A CIRCULAR CHARGED CAPILLARY

263

where the final expression is obtained by expanding the cosh term in a Taylor series for low potentials, zeψ/kB T ≪ 1. Using the above simplification, Eq. (8.128) becomes I = 2π



0

a

4π e2 z2 Dn∞ Ex ux ρf r dr + kB T



a 0



1 1+ 2



zeψ kB T

2 

r dr

(8.130)

The first term on the left hand side of Eq. (8.130) represents the current due to convection and the second term represents the current due to migration. Recognizing that for a z : z electroneutral electrolyte solution, the electric conductivity is given by Eq. (6.74) as σ∞ =

2e2 z2 Dn∞ kB T

(8.131)

and that the potential, ψ, due to the electric double layer is given by Eq. (8.101), one can rewrite the current, Eq. (8.130) as I = 2π



a

0

ux ρf r dr + Ac σ ∞ Ex Fcc

(8.132)

The term Fcc in Eq. (8.132) is given by Fcc = 1 +



zeζ kB T

2

1 2 I0 (κa)



0

1

I02 (κa · R) R dR

(8.133)

where R = r/a and Ac = π a 2 . The term Fcc is a factor that accounts for the nonelectroneutrality of the electrolyte solution due to the presence of the charged channel surface. It is equivalent to the factor derived for the slit microchannel, Fcs [see Eq. (8.58)]. Figure 8.18 depicts the variation of the function Fcc with κa for different values of dimensionless capillary surface potentials, (zeζ /kB T ). It is observed that Fcc deviates considerably from unity at low κa values, particularly for high surface potentials. Many earlier works (such as Rice and Whitehead, 1965) use Fcc = 1. One would expect that Fcc approaches unity only for κa > 5. The contribution to the total current given by the first term on the right side of Eq. (8.132) is due to the liquid bulk flow that is influenced by the pressure gradient. This is the current due to convection. The contribution of the second term on the right side of Eq. (8.132) to the total current is due to the electric conductivity of the electrolyte solution within the capillary tube. This is the current due to migration, and is often referred to as the conduction current. The total current given by Eq. (8.132) can be evaluated using expressions for ρf (r) and ux (r) as provided by Eqs. (8.102) and (8.108), respectively. Upon integration,

264

FLOW IN MICROCHANNELS

Figure 8.18. Variation of the function Fcc for a circular capillary given by Eq. (8.133) with κa for different scaled surface potentials.

Eq. (8.132) provides     2A1 2A1 2 2 2 I = − px Ac 1 − − A1 + Ex Ac σ ∞ Fcc − Ex Ac µκ 1 − κa κa      2A1 2 µκ 2 2A1 ∞ 2 1− = Ex Ac σ Fcc 1 − ∞ − A1 − px Ac 1 − (8.134) σ Fcc κa κa where = ǫζ /µ and A1 = I1 (κa) /Io (κa). The first term in Eq. (8.134) is due to the applied electric field. The second term is due to the applied pressure. As discussed in Section 8.5, one can express the current given by Eq. (8.134) in terms of a nonequilibrium thermodynamic relationship as I = L11 Ex + L12 px

(8.135)

In a similar manner, the volumetric flow rate, Q, given by Eq. (8.115) can be expressed as Q = L21 Ex + L22 px

(8.136)

Comparing the coefficients of the above non-equilibrium thermodynamic forms of the current and volumetric flow rate equations, one can satisfy the Onsager reciprocal relationship, L12 = L21 , where L12 = L21

    2A1 ǫζ Ac 2A1 =− 1− = − Ac 1 − κa µ κa

(8.137)

8.8 ELECTROOSMOTIC FLOW IN A CIRCULAR CHARGED CAPILLARY

265

For the case of px = 0, i.e., when the current and the volumetric flow rate are due to the applied electric field only, Eqs. (8.114) and (8.134) can be combined to give     L12 L21 Q 2A1 = =− ∞ 1− (8.138) f (κa, β, Fcc ) = I px =0 L11 L11 σ κa where f (κa, β, Fcc ) = and β=

1   Fcc − β 1 − 2A1 /(κa) − A21 2 µκ 2 ǫ2ζ 2κ 2 = µσ ∞ σ∞

(8.139)

(8.140)

Equation (8.138) represents the ratio of the volumetric flow rate to the applied current at zero pressure gradient. This ratio is directly related to the function f (κa, β, Fcc ). It turns out that the function only contributes to the O[(zeζ /kB T )3 ] terms in Eq. (8.138). Consequently, neglecting this term, one obtains to a leading order in zeζ /kB T ,       Q 2A1 2A1 ǫζ ≃− ∞ 1− =− ∞ 1− (8.141) I px =0 σ κa µσ κa For thin double layers, κa ≫ 1, Eq. (8.141) becomes   Q ǫζ =− ∞ =− ∞ I px =0 σ µσ

(8.142)

The ratio of (Q/I ) at zero pressure gradient is inversely proportional to the conductivity of the electrolyte solution. 8.8.4

Streaming Potential Analysis

The streaming potential is the steady potential which builds up across a capillary in the presence of an applied pressure gradient and is just sufficient to prevent any net current flow. Here, of course, we do not have an applied electric field, but we have an induced electric field. The streaming potential occurs when flow takes place in a capillary under an applied pressure gradient (Rice and Whitehead, 1965). Setting I = 0 in Eq. (8.134) and rearranging, one obtains     Ex 2A1 = ∞ 1− f (κa, β, Fcc ) (8.143) px I =0 σ κa Equation (8.143) gives the ratio of (Ex /px ) at zero electric current. Comparison between Eqs. (8.138) and (8.143) leads to       Q Ex 2A1 − = ∞ 1− f (κa, β, Fcc ) (8.144) = I px =0 px I =0 σ κa

266

FLOW IN MICROCHANNELS

which can be further simplified by retaining only the leading order term in zeζ /kB T , yielding       Q Ex 2A1 − ≃ ∞ 1− (8.145) = I px =0 px I =0 σ κa For the case of κa ≫ 1, 

Q − I



= px =0



Ex px



I =0

= ∞ = σ



ǫζ µσ ∞



(8.146)

Equation (8.144) relates electroosmosis to streaming potential flows and is in accordance with the Onsager principle of reciprocity for irreversible phenomena. Equation (8.84) for a slit microchannel is identical to Eq. (8.146) derived for a microcapillary tube.

8.8.5

Electroviscous Effect

When a liquid is forced through a narrow capillary under an applied pressure gradient (in the absence of an externally applied electric field), an induced streaming potential gradient Ex is established and the volumetric flow rate is given by Eq. (8.115). Note that the term Ex is now given by Eq. (8.144) as Ex is no longer an independent variable imposed by the experimentalist. The volumetric flow rate under a zero current condition is given by combining Eq. (8.115) with Eq. (8.143):   px Ac 2 px a 2 Ac 2A1 2 − Q= 1− f (κa, β, Fcc ) 8µ σ∞ κa    px a 2 Ac 8β 2A1 2 = 1− f (κa, β, Fcc ) 1− 8µ (κa)2 κa

(8.147)

The first term on the right side of the above expression is due to the imposed pressure gradient. It is positive in value. The second term on the right side of the above expression is due to the induced potential (streaming potential). This term is positive irrespective of the sign of the surface potential ζ . Consequently, the induced potential gives rise to a reduced volumetric flow rate for a given applied pressure. The reduction in the volumetric flow rate gives the appearance of an increased viscosity, hence the term “electroviscous effect”. The reduced rate of flow results in an apparent viscosity µa defined by Q=

px a 2 Ac 8µa

(8.148)

267

8.8 ELECTROOSMOTIC FLOW IN A CIRCULAR CHARGED CAPILLARY

Equating the volumetric flow rates provided by Eqs. (8.147) and (8.148) gives −1   µa 8β 2A1 2 = 1− 1− f (κa, β, Fcc ) µ (κa)2 κa   8β 2A1 2 ≃1+ f (κa, β, Fcc ) 1− (κa)2 κa

(8.149)

Neglecting the higher order contributions due to f (κa, β, Fcc ), Eq. (8.149) can be simplified, yielding   8β 2A1 2 µa ≃1+ (8.150) 1− µ (κa)2 κa or 8ǫ 2 ζ 2 µa = µ + ∞ 2 σ a

  2A1 2 1− κa

(8.151)

A plot of Eq. (8.150) is given in Figure 8.19. It clearly shows that for large values of β and small values of κa, µa /µ can be substantially larger than unity. Figure 8.19 also shows that packed beds having large β with κa close to 2 would not be very permeable to the electrolyte flow. For the case of κa ≫ 1, we have 1−

2A1 −→ 1 κa

Figure 8.19. Variation of normalized apparent viscosity with κa for different values of the parameter β.

268

FLOW IN MICROCHANNELS

Consequently, Eq. (8.151) provides µa ≃ µ +

8ǫ 2 ζ 2 σ ∞a2

(8.152)

This is the result of Elton (1948). The fluid acts as if it has a higher viscosity because of the additional force opposing the flow.

8.9 HIGH SURFACE POTENTIAL The study of Levine et al. (1975) extended the analysis presented here which was given by Rice and Whitehead (1965) for zeψ/kB T =  ≪ 1. This was possible through the use of the approximation given by Philip and Wooding (1970), where sinh  ≈ 

for  < 1

and sinh  ≈ 1/2 exp()

for  > 1

The equivalent equation for the axial velocity as given by Eq. (8.108) becomes   r 2  ǫ a 2 px ux (r) = − [ζ − ψ(r)] Ex 1− 4µ a µ

(8.153)

The total volumetric flow rate as given by Levine et al. (1975) is Q=

px a 2 Ac − Ac (1 − G)Ex 8µ

where G=

2 a2ζ



(8.154)

a

rψ(r) dr

(8.155)

0

The function (1 − G) is shown in Figure 8.20. The curve shown in Figure 8.16 is identical to that for s = zeζ /KB T → 0 of Figure 8.20. At κa = 1 with s → 0, Figure 8.20 indicates that (1 − G) ∼ = 0.11. This is in agreement with Figure 8.16 which shows that the term in the square brackets of Eq. (8.115) is also ∼ 0.11. However, for s = 10, (1 − G) is = 0.55. Consequently, a large deviation occurs when a high value of s is used in Rice and Whitehead’s (1965) analysis which was derived for s < 1. It should be noted, however, that the effect of high surface potential becomes less important for large κa, say κa > 100. Levine et al. (1975) also showed that large differences in (µa /µ) are involved for s > 1 when their results were compared with Rice and Whitehead’s (1965) study. The electrokinetic parameters describing electroosmosis and streaming potential for long capillaries having different cross-sectional areas, e.g., circle, ellipse, and

8.9 HIGH SURFACE POTENTIAL

269

Figure 8.20. Plot of (1 − G) versus κa for various values of the dimensionless surface potential (Levine et al., 1975).

infinite slit were treated by Anderson and Koh (1977). Based on principles similar to those outlined for the electroosmosis phenomenon, Sasidhar and Ruckenstein (1981, 1982) studied electrolyte osmosis through capillaries, and Jacazio et al. (1972) analyzed the process of electrokinetic salt rejection in hyperfiltration through porous materials. Using a similar type of analysis, salt rejection in sinusoidal capillary tubes was studied by Masliyah (1994). By the turn of the century, there has been a large increase in the number of publications dealing with flow in microchannels. This increase was prompted by the fast development of microfluidic micro-electromechanical systems (MEMS) where transport processes on the micro-scale take place in microchannel flows (NSTI, 2003; Erickson and Li, 2004; Yang, 2004; Li, 2004). Lab-on-chip devices that utilize electroosmotic flows have become a promising microfluidic technology where various molecular or colloidal entities can be fractionated, e.g., proteins, bacteria, and polymers. Some of these electroosmotic flow driven microfluidic operations in lab-on-chip devices was pioneered by Harrison and co-workers (Harrison et al., 1993; Hu et al., 1999; Cheng et al., 2001; Tang et al., 2002; Li et al., 2002; Jemere et al., 2002; Ocvirk et al., 2003). Analysis of electroosmotic flow went well beyond the basic analysis presented here. Studies dealing with electroosmotic entry flow in microchannels were conducted by Yang et al. (2001). Transient electroosmotic flows was studied by Keh and Tseng (2001). Oscillatory microchannel flows were analyzed by Yang et al. (2003) and Bhattacharyya et al. (2003). The effect of slip at the microchannel surface, where the surface is hydrophobic was investigated, among others, by Yang and Kwok (2003), Nagayama and Cheng (2004) and Yang (2004). As particulate matter move under the electroosmotic flow, they experience dispersion, i.e., spreading due to diffusion and electroosmotic flow. The effect of dispersion

270

FLOW IN MICROCHANNELS

plays an important role in particulate matter fractionation. The role of hydrodynamic dispersion on microchannel flow was studied by Martin and Guiochon (1984), Datta (1990), Griffiths and Nilson (1999), Zholkovskij et al. (2003), and Zholkovskij and Masliyah (2004). We will discuss hydrodynamic dispersion in microchannel electroosmotic flows later in this chapter. The studies mentioned above make use of the Poisson–Boltzmann equation in describing the electrokinetic transport phenomena. However, the Poisson–Boltzmann equation assumes ideal solution behavior of both the solute and the solvent. Electrokinetic models have been developed to account for finite ion size, ion hydration effects, and electric field dependent solvent dielectric constant (Gur et al., 1978a,b; Babchin and Masliyah, 1993; Lyklema, 1991; Attard, 2001). The resulting modified Poisson–Boltzmann (MPB) equation was used to show that ions of identical charge but different ionic size can yield different concentrations, dielectric constant profiles, and electric potentials within a capillary tube. The MPB equation was also used by Gur et al. (1978a) and Ravina and Gur (1985) to study electroviscous effects during capillary transport in ion-exchange membranes. The difference in the results obtained by the use of the Poisson–Boltzmann equation and its modified form is significant when the surface potential is high and the electrolyte concentration is large (Lyklema, 1991). 8.10

SURFACE CONDUCTANCE

In the analysis conducted for both the slit and the circular microchannels, attention was given to the flow region extending from the shear plane (see Figure 5.28) to the inner regions of the channel. In this context, no fluid and ionic flow were assumed to take place between the channel wall and the shear plane, namely, the Stern layer. The no-slip boundary condition and the electric surface potential in terms of zeta potential, ζ , were imposed at the shear plane. Upon imposition of a potential difference across the length of the channel, an electric current will flow. According to the analysis presented in the previous sections, the electric current is simply due to the convection and conduction in the channel within a region bounded by the shear planes. In the context of the electric double layer model used in this book, the analysis is correct. In the literature, the concept of “surface conductance” appears quite frequently. We shall attempt to clarify this concept. To begin with, we are not considering the electric conductance of the material that makes up the channel walls. In our discussion, we assume that the channel material is a perfect insulator, and therefore, no current flows within the material that forms the channel walls. Here, the conductance of the channel material is zero. Consequently, “surface conductance” has no relation to the channel material in terms of electrical conductivity. Let us now discuss surface conductance within the framework of the electric double layer. Historically, the discussion of surface conductance correction arises in context of the Smoluchowski result for the streaming potential   ǫζ Ex = = ∞ (8.156) px I =0 µσ ∞ σ

8.10

SURFACE CONDUCTANCE

271

where the effect of the electric double layer is ignored. Here, σ ∞ is the bulk conductivity of the electrolyte solution in the absence of a surface bearing a charge. As stated by Lyklema (2003), a striking feature of Eq. (8.156) is that it does not contain the radius of the microchannel. Smoluchowski, among other authors, devoted much attention to it. The fact that Eq. (8.156) does not contain the radius of the microchannel is simply because the equation is approximate, and is valid only in the limit of large κa → ∞. A more general form of the streaming potential, albeit valid only at low zeta potentials, for the case of circular microchannels is (cf., Eq. 8.145) 

Ex px



I =0

ǫζ = µσ ∞

  2A1 1− κa

(8.157)

The additional term implicitly contains the effect of the electric double layer and hence the microchannel radius. For large κa, 1 − 2A1 /(κa) becomes unity, and Eq. (8.157) becomes identical to Eq. (8.156). In terms of modern electrokinetics, there is no issue regarding incorporation of surface conductance correction as long as Eq. (8.157) is used for low zeta potentials where the effect of the electric double layer is accounted for as was done in Section 8.8.3. Before further discussion, let us consider the physical picture of current transport in the capillary corresponding to large and small values of κa. Figure 8.21 schematically shows the radial distribution of co- and counter-ion in the capillary under these conditions. For large values of κa, the electric double layer is very thin as shown in Figure 8.21(a), and only a small region near the capillary wall has any significant difference in the ion concentrations. The core of the capillary microchannel has essentially an electroneutral (n+ = n− ) electrolyte solution flowing through it. Consequently, the streaming current is zero in the capillary core. The streaming current is perceptible only when the co- and counter-ion concentrations differ, and hence, is measurable only within the electric double layer. Smoluchowski’s approximate expression for the

Figure 8.21. Schematic representation of the radial co- and counter-ion concentration distributions in a capillary for (a) large κa and (b) small κa.

272

FLOW IN MICROCHANNELS

streaming current is It = −

ǫζpx Ac µ

(8.158)

Referring back to Eq. (8.134), one immediately notes that this is an approximation of the streaming current for κa → ∞ and considers only the effect of the pressure gradient on the current. In contrast, the conduction or migration current is dominant in the core region of the capillary, and was approximated by Smoluchowski as Ic = σ ∞ Ex Ac

(8.159)

where Ac is the capillary cross-sectional area. Once again, comparing this with Eq. (8.134), one notes that the Smoluchowski expression omits the function Fcc , which accounts for the conduction current in the double layer. The rationale for this approximation is that for large κa, the double layer is vanishingly thin, and hence, its contribution to the migration current will be insignificant. Setting It = Ic , one recovers Eq. (8.156) from the above expressions. When κa is small, the electrical double layer extends well within the capillary core, and the co-ion and counter-ion concentrations differ at every radial position in the capillary (Figure 8.21b). Such a difference in the ion concentrations affects both the convective (streaming) and the migration (conduction) current through the capillary. This is clearly evident from the expression for the total current given by Eq. (8.134). The term “surface conductance” was originally introduced to incorporate the effect of a finite conductance that arises from a vanishingly thin electric double layer considered in the Smoluchowski relationship, Eq. (8.156) (Bickerman, 1940). The concept is that the total electric current due to conduction arises from two contributions. The first contribution is due to the bulk electrolyte and the second contribution is due to the surface conductance that accounts for the presence of the electric double layer. As a consequence, σ ∞ Ac in Eq. (8.159) is replaced by σ ∞ Ac + σS,dl S, yielding   σS,dl S Ic = Ex (σ ∞ Ac + σS,dl S) = σ ∞ Ac Ex 1 + ∞ (8.160) σ Ac where S is the microchannel wetted perimeter and σS,dl is the surface conductance due to the presence of the electric double layers. For a circular channel, Ac = π a 2 and S = 2π a. Introducing the correction term for σ ∞ Ac , Eq. (8.156) becomes (Hunter, 2001)   2σS,dl −1 Ex ǫζ (8.161) = 1 + px µσ ∞ aσ ∞

where σS,dl /aσ ∞ is referred to as the Dukhin number, Du (see Lyklema, 1995, 2003). For Du ≪ 1, one reverts to Eq. (8.156) where there is no dependence on the channel radius. Several studies attempted to evaluate the σS,dl term from electroosmotic measurements. The incorporation of surface conductance in the manner described above is extremely helpful when addressing electrokinetic flows in the limit of large κa → ∞.

8.10

SURFACE CONDUCTANCE

273

Comparing the conduction current term, Ac Ex σ ∞ Fcc , in Eq. (8.134) with the form given by Eq. (8.160) it is evident that for a circular capillary, Fcc = 1 +

2σS,dl aσ ∞

(8.162)

Noting that Fcc → 1 as κa > 5 from Figure 8.18, it is clearly evident that the surface conductance correction introduced in Eq. (8.161) is only manifested for small values of κa. This brings us to an important issue regarding surface conductance correction as presented by Eq. (8.161). Comparing Eqs. (8.157) and (8.161), one might erroneously infer that the term 1 − 2A1 /(κa) in Eq. (8.157) accounts for the surface conductance correction. However, this term originated from the streaming current in the detailed analysis, cf., Eq. (8.134). Smoluchowski’s expression for the streaming current, Eq. (8.158), does not contain this term. An analysis of the model presented in Section 8.8 indicates that for small κa, the dependence of the streaming current on κa is more prominent than the corresponding dependence of the conduction current on κa. The latter contribution actually does not even show up in the leading order solution for the streaming potential using the linearized Poisson–Boltzmann equation! Therefore, attempting to incorporate a surface conductance correction in the general theory of streaming potential at low κa without modifying the streaming current from the Smoluchowski form is conceptually incorrect. The surface conduction term becomes pertinent in the electrokinetic flow models only when the complete Poisson–Boltzmann equation is used in the development of the theory. Such an attempt was made by Stein et al. (2004) for a slit microchannel geometry. They used an exact analytical solution of the Poisson–Boltzmann equation (Ninham and Parsegian, 1971; Behrens and Borkovec, 1999) and developed the subsequent theoretical model for the electrokinetic flow. Their results indicate that for small κh, where h is the channel gap, the overall conductance becomes significantly different from the bulk electrolyte conductance. Furthermore, the conductance becomes independent of the channel depth and electrolyte concentration at very small κh. These observations suggest that the conductance through the electric double layer becomes the controlling parameter in the electrokinetic transport process under such conditions. The striking feature of this study is that the mathematical model developed by Stein et al. (2004) was entirely within the context of the electrical double layer theory as presented in this chapter. The only difference between their model and the one delineated in this Chapter is in the use of different forms of the Poisson– Boltzmann equation. In this context, the remarkable agreement of the model developed by Stein et al. (2004) with experiments indicates that the general theoretical construct is capable of accounting for all the pertinent physical processes associated with electrokinetic transport in microchannels, without any need to introduce the concept of surface conductance. The above discussion of the conduction current was made within the framework of the electric double layers. Consequently, “surface conductance” is confined to the ions close to the shear plane. However, there is electric current transport within the Stern layer that is not accommodated for in our analysis using the framework of the electric

274

FLOW IN MICROCHANNELS

double layer theory. Electric current transport within the Stern layer is referred to as “anomalous surface conduction”, (Hunter, 2001). Dukhin and Derjaguin (1974) introduced the term “anomalous surface conduction” to designate the tangential transfer of charge within the Stern layer. Ion movement within the layer by electromigration and diffusion is governed by the Nernst–Planck equation where the ion mobility may be different from that in the bulk solution, Saville (1998). Subsequently, Zukoski and Saville (1986) introduced the concept of a dynamic Stern layer (DSL) to model the “anomalous surface transfer”, i.e., the tangential electric current transport in the Stern layer. Such an analysis was an attempt to reconcile the differences observed between zeta potentials obtained from mobility and conductivity measurements, Mangelsdorf and White (1990). The tangential transport of ions in the hydrodynamically immobilized liquid within the Stern layer seems at first sight contradictory. As pointed out by Lyklema and Minor (1998), the situation can be compared to that of gels, e.g., gelatine in water, where a little gelling agent can immobilize water without substantially reducing the electric conductivities and diffusivities. It is quite feasible that ions actually move tangentially within the Stern layer or they simply “hop” to the diffuse layer and then return to the Stern layer, and in so doing, move along the surface of the capillary. Details of the ion transport within the Stern layer still have to be resolved. If one is to include the Stern layer surface conductance, then Eq. (8.161) takes the form   ǫζ 2(σS,dl + σss ) −1 Ex = (8.163) 1 + px µσ ∞ aσ ∞ where σss is surface conductance due to the mobile ions in the Stern layer. When anomalous surface conduction occurs, i.e., σss
= 0, it leads to an under-estimation of the zeta potential, ζ , using electrophoresis measurements, as is evident from Eq. (8.163).

8.11

SOLUTE DISPERSION IN MICROCHANNELS

In our analysis of electroosmotic flow in narrow channels thus far, we have neglected solute dispersion effects. In this section, we will briefly describe the phenomena of hydrodynamic and diffusional dispersion (Taylor, 1953; Aris, 1956), and assess their importance in electroosmotic flow. When a fluid flows in a narrow channel, for instance, in the case of a onedimensional parabolic pressure driven flow in a cylindrical capillary, the axial velocity of the fluid varies with radial position. The velocity is zero at the channel wall (owing to the no-slip condition) and attains a maximum value at the center of the channel. If we consider a solute being convected by this fluid, naturally, the solute molecules at different radial positions in the channel will travel with different convective velocities. We also know that molecules undergo random thermal agitation, or diffusion, which leads to a diffusive flux of the solute from high to low concentration regions.

8.11

SOLUTE DISPERSION IN MICROCHANNELS

275

Figure 8.22. The concept of dispersion in a microchannel flow. An initially uniform solute concentration band translates with the flow and broadens due to diffusion and hydrodynamic effects.

Due to this diffusive motion, the solute molecules can diffuse from one flow streamline to another neighboring flow streamline having a different velocity. The net effect of these two phenomena is a broadening of an initially uniform band of solute. This process is generally termed dispersion. In other words, diffusion, coupled with a spatially non-uniform velocity field, tends to enhance the transport of solutes, causing an enhanced mixing of the solutes. Figure 8.22 schematically depicts the process of dispersion. The dispersion of solutes plays an important role in many technological processes. For example, dispersion is always detrimental in separation and fractionation processes, since it leads to the mixing of those substances that are being separated from each other. Conversely, some chemical technologies require substances to be homogenized within a certain volume and dispersion can provide the homogenization. In the case of transport through a straight channel, dispersion occurs due to (i) longitudinal diffusion of the solute (diffusional dispersion) and (ii) non-uniformity in the liquid velocity within a channel cross-section (hydrodynamic dispersion). Being a consequence of the thermal motion of molecules, the diffusional dispersion is unavoidable.As for the hydrodynamic dispersion, it can be substantially reduced by employing a hydrodynamic flow with a nearly uniform velocity distribution within the channel cross-section. For example, a substantial reduction of dispersion is observed when, for transportation of a solute, electroosmosis is used instead of pressure driven flow. As it was shown earlier in this chapter, when κa ≫ 1, the electroosmotic velocity is nearly uniform over the cross-section except at the thin interfacial electric double layer region. In contrast, the parabolic type pressure driven (Poiseuille) flow is strongly non-uniform. Consequently, electroosmosis produces much weaker dispersion than the Poiseuille flow. Due to such a property, electroosmotic flow of a liquid through a capillary is widely employed in microfluidic devices as a means for solute transport. 8.11.1 Diffusional and Hydrodynamic Dispersion To understand the mechanism of purely diffusional dispersion, we will consider the time evolution of a solute concentration band in a uniform flow as shown in

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FLOW IN MICROCHANNELS

Figure 8.23. Dispersion of a concentration band for (a) a uniform velocity field but zero diffusion, (b) zero velocity but non-zero diffusion, and (c) uniform velocity and non-zero diffusion.

Figure 8.23. Each of the solute molecules participates in a translational motion with the liquid flow and in a random thermal motion that results in diffusion. It is first noted that when the thermal motion is vanishingly weak (negligible diffusion), the solute molecules have a common and time independent velocity in the uniform flow field. Consequently, the solute band is translated as a solid body i.e., without any deformation [Figure 8.23(a)]. To analyze the role of thermal motion, consider a reference system linked to the concentration band in the liquid which moves with velocity ux (Figure 8.23b). In this reference system, the molecules participate solely in thermal motion, giving rise to the solute diffusion fluxes that are directed toward the lower solute concentrations (outward from the concentration band). Thus, the longitudinal diffusion leads to a permanent transport of the molecules out from the concentration band. Combining the translational (convective) and the diffusive effects, one can observe a superposition of the behavior shown in Figure 8.23(a) and (b): the band is translated with the liquid velocity, ux , and is broadened [Figure 8.23(c)]. To understand the mechanism of hydrodynamic dispersion, consider a solute band in a flow with a non-uniform velocity distribution over a channel cross-section (Figure 8.24). For the limiting case of vanishingly slow diffusion, the solute molecules participate in a purely convective motion with different velocities. Owing to the difference in velocities, different parts of an initial solute band are transported to different distances. As a result, the band is deformed [Figure 8.24(a)], and its longitudinal dimension, Wh , becomes longer than that of the initial band, W0 . At a given moment, and for a given channel geometry, such a longitudinal hydrodynamic spreading becomes stronger with increasing mean cross-sectional velocity.

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Figure 8.24. Dispersion of concentration band (a) for non-uniform velocity in a channel (purely hydrodynamic dispersion), and (b) for the general case of convection in a non-uniform velocity field and diffusion.

Longitudinal and transverse diffusion, when superimposed on the hydrodynamic dispersion, lead to qualitatively different results. Longitudinal diffusion manifests itself as diffusional dispersion. Basically, its role remains the same as for the case of a uniform flow, i.e., it amounts to additional broadening of the band. As for transverse diffusion, it results in decreasing the hydrodynamic dispersion. The non-uniformity of a velocity distribution leads to the non-uniformity of the concentration distribution within channel cross-sections [Figure 8.24(a)] and, consequently, to transverse diffusion fluxes. Due to the transverse diffusion fluxes, the solute concentration is levelled within any cross-section, and, for sufficiently narrow channels, becomes nearly (but not perfectly) constant. As a result, the solute concentration only varies in the longitudinal direction, thereby forming a longitudinal concentration profile [Figure 8.24(b)]. Such a profile is wider than the initial concentration band (W > W0 ). However, it is always narrower than the longitudinal distribution which corresponds to the hydrodynamically deformed band shown in Figure 8.24(a) (W < Wh ). To explain this fact, we will consider three cross-sections A, B, and C, shown in Figure 8.25. Within cross-section A, which is chosen in the rear part of the deformed band, the solute molecules diffuse from a stagnant zone of the band into the region where the liquid has a high velocity (Figure 8.25). Consequently, these molecules move with the liquid toward the center of the band (cross-section C). Simultaneously, within cross-section B, which is chosen at the leading part of the band, the molecules diffuse from the fast moving part of the band into the slower moving zone of the liquid. These molecules also approach cross-section C. Thus, due to transverse diffusion, the mean cross-sectional concentration decreases at the ends (cross-sections A and B) and increases in the middle of the band (cross-section C). This causes the

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Figure 8.25. Narrowing of longitudinal concentration band due to transverse diffusion.

longitudinal distribution to become narrower (Figure 8.25). Consequently, the band width W satisfies the inequality W0 < W < Wh (Figure 8.24). The above discussion demonstrates that transverse diffusion strongly affects hydrodynamic dispersion – a stronger transverse diffusion leading to a weaker hydrodynamic dispersion. It is remarkable that, for the limiting case of infinitely fast diffusion, when the transverse concentration distribution has time to become perfectly uniform within a cross-section, the hydrodynamic dispersion does not manifest itself. The main properties of diffusional and hydrodynamic dispersion are summarized below: Diffusional Dispersion • is a consequence of the thermal motion of molecules; • exists in the case of a uniform flow; • becomes stronger for solutes having larger diffusion coefficients. Hydrodynamic Dispersion • • • •

is a consequence of the non-uniformity in liquid velocity within a cross-section; does not exist in the case of a uniform flow; becomes stronger with increasing mean cross-sectional velocities; becomes weaker for solutes having higher diffusion coefficients.

In the general case, one deals with a combination of both dispersion types. 8.11.2 Convective-Diffusional Transport Through Channels To analyze the dispersion behavior of solutes, one needs to track the spatio-temporal evolution of the solute concentration band flowing through a channel. The time

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279

Figure 8.26. Microchannel geometry and coordinate framework employed to analyze hydrodynamic and diffusional dispersion in a one-dimensional flow. (a) Left: side view, (b) right: cross-sectional view.

dependent spatial distribution of the local solute concentration, c = c(x, y, z, t), can be expressed in terms of the convective diffusion equation as ∂c = D∇ 2 c − u · ∇c ∂t

(8.164)

where D is the diffusion coefficient of the solute and the velocity field u is provided from an appropriate solution of the Navier–Stokes and continuity equations. The general geometry under consideration is depicted in Figure 8.26. Considering a steady flow in a long straight channel and neglecting end effects, we make the following simplifying assumptions regarding the flow structure: (i) the velocity is directed along the x-axis, and (ii) at all cross-sections, the velocity distributions are equal, i.e., the velocity distribution does not depend on the longitudinal (axial) coordinate, x. Consequently, one can represent the velocity as u = ix ux (y, z)

(8.165)

where ix is the unit vector attributed to the x-axis. Combining Eqs. (8.164) and (8.165) one obtains ∂c ∂ 2c ∂c 2 = D 2 + D∇yz c − ux (y, z) ∂t ∂x ∂x

(8.166)

where the 2D Laplace operator is given by 2 ∇yz =

∂2 ∂2 + ∂y 2 ∂z2

One can now define a cross-sectional average concentration as 1 c(x, ¯ t) = A



A

c(x, y, z, t) dA

(8.167)

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where A is the cross-sectional area and dA = dy dz. In a similar manner, one can also define a cross-sectional average velocity u¯ x . Furthermore, the local variations in concentration and velocity from their cross-sectional average values can be represented as c = c¯ + δc

and ux = u¯ x + δux

(8.168)

Employing Eqs. (8.167) and (8.168), and applying a set of averaging rules, the convective–diffusion equation (8.166) can be written as ∂ c¯ ∂ c¯ ∂ 2 c¯ ∂δux δc = D 2 − u¯ x − ∂t ∂x ∂x ∂x

(8.169)

The last term in Eq. (8.169) becomes zero when the velocity and/or the concentration are uniformly distributed over the cross-section (δux = 0 and/or δc = 0, respectively). According to the earlier discussion, for these cases, one can expect purely diffusional dispersion of the solute band. Consequently, omitting the final term Eq. (8.169) describes the regime of purely diffusional dispersion. 8.11.2.1 Dispersion in Uniform Flow Assuming that the flow is uniform (δux = 0), Eq. (8.169) simplifies to ∂ c¯ ∂ c¯ ∂ 2 c¯ = D 2 − u¯ x ∂t ∂x ∂x

(8.170)

The cross-sectional mean concentration should satisfy the boundary conditions at infinity given by c(x ¯ = −∞) = c(x ¯ = ∞) = 0

(8.171)

and the initial condition (at t = 0) given by c(x, ¯ t = 0) = c¯0 (x)

(8.172)

On the right hand side of Eq. (8.172), the function c¯0 (x) is the longitudinal distribution of the cross-sectional mean concentration at t = 0. Due to the incompressibility of liquid, for a straight channel, the mean crosssectional velocity u¯ x does not depend on the longitudinal coordinate, x. Thus, the homogeneous partial differential equation (8.170) depends on constant coefficients D and u¯ x . Applying the initial and boundary conditions, Eq. (8.170) can be solved, yielding, c(x, ¯ t) =

1 2(πDt)1/2

  (x − x ′ − u¯ x t)2 c¯0 (x ′ ) exp − dx ′ 4Dt −∞



∞

(8.173)

Equation (8.173) can be used to predict the longitudinal concentration profile at any time, provided the concentration profile c¯0 (x) at t = 0 is a known function.

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To assess the dispersion behavior in a quantitative manner, several parameters can be defined. The most common geometrical and kinematic parameters quantifying the dispersion behavior are, the mass center coordinate, the variance and band width, the velocity of mass center, and the dispersion coefficient. These terms are defined below: Mass Center Coordinate: This quantity provides the instantaneous location of the center of mass of the solute band. When the solute molecules have a common molecular weight, the mass center coordinate can be represented as ∞

xc (t) = −∞ ∞

−∞

x c(x, ¯ t)dx c(x, ¯ 0)dx

(8.174)

The above function bears information about the translational movement of the band. Employing the concentration distribution given by Eq. (8.173) in Eq. (8.174), one observes xc = xc0 + u¯ x t

(8.175)

where xc0 denotes the mass center coordinate of the initial solute band at t = 0. This implies that the center of mass of the concentration band translates as a linear function of time in the case of purely diffusional dispersion in a uniform velocity field. Variance and Band Width: Dispersion is often described using the parameter known as the band width, W . It should be noted that, usually, a band does not have sharp external borders. A universal approach to describing the band width of dispersion is to use the variance, σ , which is the root mean square deviation of the molecule coordinate, x, from the mass center coordinate, xc . 2

σ (t) =

∞

− xc (t)]2 c(x, ¯ t)dx ∞ ¯ 0)dx −∞ c(x,

−∞ [x

(8.176)

The band width is expressed in terms of the variance as W = βσ

(8.177)

where the dimensionless factor β depends on the shape of the concentration profile. For a Gaussian distribution β = 4, whereas for a rectangular distribution β = 6. For the present case, using the concentration profile from Eq. (8.173) in Eq. (8.176), we obtain σ 2 (t) = σ02 + 2Dt

(8.178)

where the variance of the initial concentration distribution is denoted by σ0 . The mass center coordinate, xc , and the variance, σ (or the band width, W ), characterize the geometry of the solute concentration band at a given instant. These geometrical parameters change with time. To address the rate of these changes, a set of relevant kinematical parameters is usually employed.

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Velocity of Mass Center: The mass center velocity, uc , is defined as uc =

dxc dt

(8.179)

For the uniform velocity field, the mass center velocity is simply given by uc = u¯ x

(8.180)

Dispersion Coefficient: The dispersion coefficient, K, describes the rate of the change in the squared variance with time. It is defined as K=

1 dσ 2 (t) 2 dt

(8.181)

Employing Eq. (8.178) in Eq. (8.181) yields K=D

(8.182)

Thus, for purely diffusional dispersion in a uniform velocity field, the dispersion coefficient is equal to the diffusion coefficient of the solute. 8.11.2.2 Dispersion in Non-Uniform Flow: Taylor–Aris Theory We now consider the transport of the solute concentration band in a flow of the general type. Accordingly, we will deal with a non-uniform velocity distribution, for which, the last term of Eq. (8.169), ∂δux δc/∂x, takes a non-zero value (except for the case of a perfectly uniform solute concentration, δc = 0). In contrast to the case for uniform velocity field, Eq. (8.170), where the only unknown was the cross-sectional average solute concentration, c(x, ¯ t), solution of the general transport equation, Eq. (8.169), involves determination of two unknowns, namely, c(x, ¯ t) and δc(x, y, z, t). Therefore, it is necessary to obtain an additional equation, which, together with Eq. (8.169), would form an equation set for obtaining the two unknown functions. This additional equation can be derived by substituting Eq. (8.168) in Eq. (8.166). This second partial differential equation pertains to the concentration fluctuation, δc(x, y, z, t). After assigning appropriate boundary conditions for the second equation, one can solve the two equations employing a perturbation approach. For details of the approach, one is referred to the original works of Taylor (Taylor, 1953) and Aris (Aris, 1956; Aris, 1959). Their approach leads to an equation for the spatio-temporal evolution of the cross-sectional average concentration of the form   ∂ c¯ u¯ 2x H∗2 ∂ 2 c¯ ∂ c¯ = D+ (8.183) − u¯ x 2 ∂t 16D ∂x ∂x where H∗ is a length-scale parameter, often referred to as the minimum plate height, that depends on the cross-sectional geometry of the channel. Thus, for an arbitrary distribution of the flow velocity within a channel cross-section, the behavior of the

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283

mean cross-sectional concentration c(x, ¯ t) is described by Eq.(8.183) subject to the boundary conditions, Eq. (8.171), and the initial condition, Eq. (8.172). Let us now compare the equation describing the behavior of the concentration band derived for a uniform flow, namely Eq. (8.170), with Eq. (8.183), which is derived for an arbitrary velocity distribution. Remarkably, the equations have a common structure and differ from each other solely due to the difference in the coefficients of the second derivative term. In Eq. (8.170) it is the diffusion coefficient D, whereas, in Eq. (8.183), it is a quantity, (D + u¯ 2x H∗2 /16D), which depends on the mean crosssectional velocity, u¯ x . According to Eq. (8.182), for uniform flow, when purely diffusional dispersion is observed, the dispersion coefficient K is equal to the diffusion coefficient D. In contrast, for a flow of general type, the dispersion coefficient is given by (cf., Eq. 8.183),

K =D 1+



u¯ x H∗ 4D

2 

(8.184)

Equation (8.184) yields the dispersion coefficient for a flow of the general type when both diffusional and hydrodynamic dispersions manifest themselves. Such an expression for the dispersion coefficient, in a slightly different form, was obtained by Taylor (1953) who considered Poiseuille flow √ through a channel with a circular cross-section. Accordingly, Taylor used H∗ = a/ 3, where a is the radius of the channel in Eq. (8.183). Later, the Taylor theory was generalized by Aris (1956) to address dispersion for an arbitrary channel cross-section. The approach of Aris leads to Eq. (8.184).

8.11.3 Dispersion in a Slit Microchannel We now assess the dispersion of solutes in a slit microchannel. Two types of flow will be considered, namely pressure driven Poiseuille type flow and electroosmotic flow. It should be noted at the outset, that for the given geometry, the variables (concentration and velocity) will be independent of the z-coordinate. 8.11.3.1 Pressure Driven Flow For the case of purely pressure driven flow, the velocity distribution, ux (y), is given by the expression ux (y) =

  y 2  3 u¯ x 1 − 2 h

(8.185)

where h is the channel height. It can be shown that the minimum plate height, H∗ , is related to the channel height by H∗2 =

32 2 h 105

(8.186)

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FLOW IN MICROCHANNELS

Using the above relationship, one can now easily obtain the dispersion coefficient for this type of flow    u¯ x h 2 2 (8.187) K =D 1+ 105 D This expression for the dispersion coefficient was obtained by Aris (1959). 8.11.3.2 Electroosmotic Flow For purely electroosmotic flow, which is observed in absence of the longitudinal pressure gradients, the velocity distribution, ux (y), is given by the second term of Eq. (8.30) ux (y) = −

  cosh(κy) ǫEx ζ 1− µ cosh(κh)

(8.188)

The cross-sectional mean velocity, obtained from Eq. (8.188), is   ǫEx ζ tanh(κh) u¯ x = − 1− µ (κh)

(8.189)

Griffiths and Nilson (1999) used these expressions to obtain for the minimum plate height, H∗ , the following expression H∗2 =

4 f (κh) 3κ 2

(8.190)

where f (κh) =

4[6 + (κh)2 ] sinh2 (κh) − 9(κh) sinh(2κh) − 6(κh)2 [(κh) cosh(κh) − sinh(κh)]2

(8.191)

The variation of the normalized minimum plate height, κH∗ , with the scaled channel width, κh, is depicted in Figure 8.27. Now, substituting the average velocity and the minimum plate height in Eq. (8.184), the dispersion coefficient for purely electroosmotic flow in a slit microchannel can be determined as

     1 ǫEx ζ 2 tanh(κh) 2 K =D 1+ f (κh) (8.192) 1− 12 µκD κh For κh ≪ 1, the function f (κh) in Eq. (8.191) can be approximated as f (κh) =

8 (κh)2 + O[(κh)4 ] 35

(8.193)

Substituting, Eq. (8.193) in Eq. (8.190), and retaining the leading term, we arrive at the expression given by Eq. (8.186). Thus, for electroosmotic flow through a channel with small κh, the asymptotic expression for the minimum plate height is the same

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285

as that obtained for pressure driven flow. This happens because for κa ≪ 1, the electroosmotic velocity profile becomes similar to the pressure driven flow velocity profile. For large values of the parameter κh ≫ 1, the electroosmotic velocity profile becomes uniform (independent of y), given by the Helmholtz–Smoluchowski expression ux = −

ǫEx ζ µ

(8.194)

In this case of κh ≫ 1, the expression for the function f (κh) becomes lim f (κh) = 4

(8.195)

κh→∞

and the minimum plate height, H∗ , is given by lim H∗2 =

κh→∞

16 3κ 2

(8.196)

This asymptotic value is depicted in Figure 8.27. Hence, for κh →√∞, the normalized minimum plate height, κH∗ , approaches the constant value of 4/ 3. The dispersion coefficient for this type of flow simply becomes

K =D 1+



ǫEx ζ √ 3µκD

2 

(8.197)

Figure 8.27. Variation of the scaled minimum plate height, κH∗ , with the scaled width of the slit microchannel, κh.

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FLOW IN MICROCHANNELS

In the limit κ → ∞, one recovers from Eq. (8.197) the purely diffusional dispersion behavior for the uniform velocity profile K=D

(8.198)

To summarize, the influence of hydrodynamic and diffusional dispersion is perceptible in electroosmotic flows in channels where κh ≪ 1. In this case, however, the dispersion behavior is similar to that observed for a purely pressure driven parabolic velocity profile. For large values of κh, i.e., κh ≫ 1, the hydrodynamic dispersion has no influence on the solute transport, and the dispersion process is solely dictated by diffusion.

8.12 a Ac c c¯ D Dk Du e Ex h H∗ i ix Ix Ic It Io I1 I kB K n+ n_ nk n∞ p

NOMENCLATURE capillary tube radius, m cross-sectional area of slit microchannel per unit width, m2 /m capillary tube cross-sectional area, m2 solute concentration cross-sectional average solute concentration solute diffusion coefficient, m2 /s diffusivity of k th ionic species, m2 /s Dukhin number elementary charge, C electric field in axial direction, V/m half width of a slit microchannel, m minimum plate height, m current density vector, A/m2 component of the current density vector along axial flow direction, A/m2 total current per unit width for slit microchannel, A/m total current for capillary microchannel, A conduction current per unit width for slit microchannel, A/m conduction current for capillary microchannel, A transport or streaming current per unit width for slit microchannel, A/m transport current for capillary microchannel, A zero-order modified Bessel function first-order modified Bessel function total current Boltzmann constant, J/K dispersion coefficient, m2 /s ionic number concentration of the cations, m−3 ionic number concentration of the anions, m−3 ionic number concentration of k th species, m−3 ionic number concentration in the bulk solution, m−3 fluid pressure, Pa

8.13

px Q Qel

r T u uc ux u¯ VFR x xc z zk

PROBLEMS

287

negative pressure gradient, −dp/dx, Pa/m volumetric flow rate per unit width in a slit microchannel, m3 /ms total volumetric flow rate in capillary microchannel, m3 /s volumetric flow rate per unit width due to electrical field in a slit microchannel, m3 /ms volumetric flow rate due to electrical field in capillary microchannel, m3 /s radial coordinates, m absolute temperature, K fluid velocity vector, m/s velocity of mass center, m/s local axial fluid velocity, m/s cross-sectional average fluid velocity, m/s volumetric flow ratio axial coordinate, m center of mass coordinate of a solute concentration band absolute value of a (z : z) electrolyte solution valency valency of the k th species

Greek Symbols β ǫ κ µ µa ρf σ∞ σS,dl σ φ φo ψ s  ζ

8.13

2 µκ 2 /σ ∞ dielectric permittivity of solvent, C/Vm inverse Debye length, m−1 electrolyte solution viscosity, Pa s apparent electrolyte solution viscosity, Pa s free charge density, C/m3 electrolyte solution conductivity, S/m or A/Vm surface conductance, S or A/V variance of the solute concentration band, m2 total potential, V total potential at x = 0, V potential due to electric double layer, V scaled surface potential (at the shear plane) zeζ /kB T dimensionless potential, zeψ/kB T zeta potential of surface, V ǫζ /µ

PROBLEMS

8.1. For the problem of electroosmotic flow in a slit microchannel discussed in Section 8.2, the boundary condition given at the channel mid-plane, Eq. (8.16-b), is to be replaced by ψ =0

at y = 0

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FLOW IN MICROCHANNELS

This boundary condition has been used quite frequently in the literature. Clearly, it is only valid for large κh. Such a boundary condition leads to ψ =ζ

sinh(κy) sinh(κh)

Derive an expression for −ux,el /( Ex ) and compare your solution with the corresponding Eq. (8.32) and Figure 8.4. 8.2. In Section 8.2.2, we assumed that for a slit microchannel, the pressure gradient is fairly independent of the transverse y-direction. Let us examine this approximation. (a) To avoid the assumption of a narrow channel where the hydrostatic head is not important, let us deal with the dynamic pressure, defined as p + ρgy, where g is the acceleration due to gravity. Show that the dynamic pressure gradient for a (z : z) electrolyte using the Debye–Hückel approximation is given by cosh(κy) sinh(κy) 1 ∂(p + ρgy) = ǫκ 2 ζ 2 ∂y cosh2 (κh) (b) Making use of the identities sinh(2x) = 2 sinh(x) cosh(x) and cosh(2x) = 1 + 2 sinh2 (x) show that the dynamic pressure is given by p + ρgy =



ǫκ 2 ζ 2 4



 1 + 2 sinh2 (κy) + G(x) cosh2 (κh)

Show that for κh → 0, the term in the square brackets become unity and the dynamic pressure is constant across the channel width for a given axial location. Show that for κh → ∞, the dynamic pressure is given by p + ρgy =

ǫκ 2 ζ 2 2



sinh(κy) cosh(κh)

2

+ G(x)

The term in the square brackets in the above equation is zero except very close to the channel wall, indicating the dynamic pressure is fairly constant across the channel width. 8.3. The factor Fcc is given as Fcc = 2



0

1

  I0 (κaR) cosh s · RdR I0 (κa)

8.13

PROBLEMS

289

where R = r/a and s = zeζ /kB T . The modified Bessel function of the first kind is given by an integral expression In (x) = √

 x n π/2 2  cosh(x sin θ )(cos θ )2n dθ 2 π Ŵ n + 21 0

where Ŵ is the Gamma function and n is the order of the modified Bessel function. Evaluate Fcc for s = 1, 1.5, and 2. Compare with the integral Fcs for the case of a slit microchannel. 8.4. Solve the electroosmotic flow problem for the slit microchannel where symmetry at y = 0 is not assumed. Here the surface potentials are given as ψ = ζ1

at y = h

and ψ = ζ2

at y = −h

(a) Obtain an expression for ψ(y) that is equivalent to Eq. (8.17). (b) Obtain an expression for ux (y) that is equivalent to Eq. (8.30). (c) Compare the volumetric flow rates, Qel , for the cases (i) ζ1 = ζ2 = 1.5 and (ii) ζ1 = 2 and ζ2 = 1.0 for κh = 1, 5, and 50. Assume Debye–Hückel assumption to hold. 8.5. A circular capillary has an electric surface potential of −0.05 V, a diameter of 10 µm and a length of 3 cm. The capillary tube is connected to two reservoirs containing 0.01 M CaSO4 . Electrodes are immersed in the two reservoirs and the applied potential difference is 100 V. Assume that the solution viscosity is 10−3 Pa.s at 20◦ C. (a) What is the volumetric flow rate of the electrolyte solution? (b) What would be the required pressure drop to affect the volumetric flow rate in part (a) without an electric field? 8.6. Consider water at 20◦ C containing 0.001 M NaCl flowing in a slit microchannel under the influence of an external electric field of 1000 V/m. The channel wall zeta potential is 50 mV. The material of the channel wall is not conductive. (a) Plot the volumetric flow rate per unit area (m3 /m2 s) versus channel half height, h (nm). (b) Evaluate the pressure gradient required to achieve the electroosmotic flow rate for a given channel half height. (c) Evaluate the current per unit area as a function of channel half height. Comment on the plots obtained in parts (a) to (c).

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FLOW IN MICROCHANNELS

8.7. It is of interest to compare dispersion between electroosmotic and pressure driven flows for the same average velocity and dimensionless slit channel half height, κh. (a) Show that (K/D − 1)pressure 105f (κh) = (K/D − 1)electroosmotic 24(κh)2 The function f (κh) is that given by Eq. (8.191). (b) Plot the ratio (K/D − 1)pressure /(K/D − 1)electroosmotic against κh. (c) From the plot under (b) one observes that the dispersion ratio is nearly unity when κh < 1. Would there be any advantage in using a channel with κh < 1 to minimize hydrodynamic dispersion under electroosmotic flow? (d) What is the value of the dispersion ratio for κh = 100? Can one minimize hydrodynamic dispersion for κh = 100 under electroosmotic flow? (e) Why is the dispersion ratio so different at small and large values of κh? (f) Plot the variation of the dispersion ratio with h for water flowing in slit channels with different heights at 20◦ C. The electrolyte is CaCl2 at 0.001 M. 8.14

REFERENCES

Anderson, J. L., and Koh, W. H., Electrokinetic parameters for capillaries of different geometries, J. Colloid Interface Sci., 59, 149–158, (1977). Aris, G., On the dispersion of a solute in fluid flowing through a tube, Proc. Roy. Soc. Lond., 235A, 67–77, (1956). Aris, G., On the dispersion of a solute by diffusion, convection and exchange between phases, Proc. Roy. Soc. Lond., 252A, 538–550, (1959). Attard, P., Recent advances in the electric double layer in colloid science, Curr. Opin. Colloid Interface Sci., 6, 366–371, (2001). Babchin, A. J., and Masliyah, J.H., Modified Nernst–Planck equation for hydration effects, J. Colloid Interface Sci., 160, 258–259, (1993). Behrens, S. H., and Borkovec, M., Exact Poisson–Boltzmann solution for the interaction of dissimilar charge-regulating surfaces, Phys. Rev. E., 60, 7040–7048, (1999). Bhattacharyya, A., Masliyah, J. H., and Yang, J., Oscillating laminar electrokinetic flow in infinitely extended circular microchannels, J. Colloid Interface Sci., 261, 12–20, (2003). Bickerman, J. J., Electrokinetic equations and surface conductance. A survey of the diffuse double layer theory of colloidal solutions, Trans. Faraday Soc., 36, 154–160, (1940). Burgreen, D., and Nakache, F. R., Electrokinetic flow in ultrafine capillary slits, J. Phys. Chem., 68, 1084, (1964). Cheng, S. B., Skinner, C., Taylor, J., Attiya, S., Lee, W. E., Picelli, G., and Harrison, D. J., Development of a multi-channel microfluidic analysis system employing affinity capillary electrophoresis for immunoassay, Anal. Chem., 73, 1472–1479, (2001). Datta, R., and Kotamarthi, V. R., Electrokinetic dispersion in capillary electrophoresis, AIChE J., 36, 916–926, (1990).

8.14

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Dukhin, S. S., and Derjaguin, B. V., Electrokinetic phenomena, in Surface and Colloid Science, vol. 7, E. Matijevic (Ed.), Wiley, (1974). Elton, G. A. H., Electroviscosity I. The flow of liquids between surfaces in close proximity, Proc. Roy. Soc. Lond., 194A, 259–274, (1948). Erickson, D., Li, D. Q., and Werner, C., An improved method of determining the zeta-potential and surface conductance, J. Colloid Interface Sci., 232, 186-197, (2000). Erickson, D., and Li, D. Q., Integrated microfluidic devices, Analytica Chim. Acta, 507, 11–26, (2004). Griffiths, S. K., and Nilson, R. H., Hydrodynamic dispersion of a neutral non-reacting solute in electroosmotic flow, Anal. Chem., 71, 5522–5529, (1999). Gur, Y., Ravina, I., and Babchin, A. J., On the electrical double layer theory. I. Numerical method for solving a generalized Poisson–Boltzmann equation, J. Colloid Interface Sci., 64, 326–332, (1978a). Gur, Y., Ravina, I., and Babchin, A. J., On the electrical double layer theory. II. The Poisson– Boltzmann equation including hydration forces, J. Colloid Interface Sci., 64, 333–341, (1978b). Guzman-Garcia, A. G., Pintauro, P. N., Verbrugge, M. W., and Hill, R. F., Development of a space-charge transport model for ion-exchange membranes, AIChE J., 36, 1061–1074, (1990). Harrison, D. J., Fluri, K., Seiler, K., Fan, Z. H., Effenhauser, C. S., and Manz, A., Micromachining a miniaturized capillary electrophoresis-based chemical-analysis system on a chip, Science, 261, 895–897, (1993). Hiemenz, P. C., and Rajagopalan, R., Principles of Colloid and Surface Chemistry, 3rd ed., Marcel Dekker, New York, (1997). Hu, L. G., Harrison, D. J., and Masliyah, J. H., Numerical model of electrokinetic flow for capillary electrophoresis, J. Colloid Interface Sci., 215, 300–312, (1999). Hunter, R. J., Zeta Potential in Colloid Science, Academic Press, London, (1981). Hunter, R. J., Foundations of Colloid Science, 2nd ed., Oxford University Press, Oxford, (2001). Jacazio, A., Probstein, R. F., Sonin, A. A., and Ying, E., Electrokinectic salt rejection in hyperfiltraion through porous material. Theory and experiment, J. Phys. Chem., 76, 4015–4023, (1972). Jemere, A. B., Oleschuk, R., Ouchen, F., Fajuyigbe, F., and Harrison, D. J., An integrated solid phase extraction system for sub-pico molar detection, Electrophoresis, 23, 3537–3544, (2002). Keh, H. J., and Tseng, H. C., Transient electrokinetic flow in fine capillaries, J. Colloid Interface Sci., 242, 450–459, (2001). Levine, S., Marriott, J. R., Neale, G., and Epstein, N., Theory of electrokinetic flow in fine cylindrical capillaries at high zeta-potentials, J. Colloid Interface Sci., 52, 136–149, (1975). Li, D. Q., Electrokinetics in Microfluidics, Elsevier, Amsterdam, (2004). Li, J., LeRiche, T., Tremblay, T. L., Wang, C., Bonneil, E., Harrison, D. J., and Thibault, P., Application of microfluidic devices to proteomics research, Mol. and Cell. Proteomics, 1.2, 157–168, (2002). Lyklema, J., Fundamentals of Interface and Colloid Science, vol. 1, Academic Press, London, (1991).

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Lyklema, J., Fundamentals of Colloid and Interface Science, vol. II, Academic Press, London, (1995). Lyklema, J., and Minor, M., On surface conduction and its role in electrokinetics, Colloids Surf. A, 140, 33–41, (1998). Lyklema, J., Electrokinetics after Smoluchowski, Colloids Surf. A, 222, 5–14, (2003). Mangelsdorf, C. S., and White, L. R., Effects of Stern-layer conductance on electrokinetic transport properties of colloidal particles, J. Chem. Soc. Faraday Trans., 86, 2859–2870, (1990). Martin, M., and Guiochon, G., The axial-dispersion in open-tubular capillary liquidchromatography with electroosmotic flow, Anal. Chem., 56, 614–620, (1984). Masliyah, J. H., Salt rejection in a sinusoidal capillary tube, J. Colloid Interface Sci., 166, 383–393, (1994). Mazur, P., and Overbeek, J. Th. G., On electro-osmosis and streaming-potentials in diaphragms .2. general quantitative relationship between electro-kinetic effects, J. Roy. Netherlands Chem. Soc., 70, 83–91, (1951). Nagayama, G., and Cheng, P., Effects of interface wettability on microscale flow by molecular dynamics simulation, Int. J. Heat Mass Transfer, 47, 501–513, (2004). Nano Science and Technology Institute (NSTI), Technical Proceedings of the 2003 Nanotechnology Conference and Trade Show, NSTI, Cambridge, MA, (2003). Ninham, B. W., and Parsegian, V. A., Electrostatic potential between surfaces bearing ionizable surface groups in ionic equilibrium with physiologic saline solution, J. Theor. Biol., 31, 405–428, (1971). Ocvirk, G., Salimi-Moosavi, H., Szarka, R. J., Arriaga, E. A., Andersson, P. E., Smith, R., Dovichi, N. J., and Harrison, D. J., β-galactosidase assays of single cell lysates on a microchip: A complementary method for enzymatic analysis of single cells, IEEE Spl. Iss. on Biomedical Applications for MEMS and Microfluidics, (2003). Philip, J. R., and Wooding, R. A., Solution of the Poisson–Boltzmann equation about a cylindrical particle, J. Chem. Phys., 52, 953–959, (1970). Pintauro, P. N., and Verbrugge, M. W., The electric-potential profile in ion-exchange membranes, J. Membrane Sci., 44, 197–212, (1989). Ravina, I., and Gur, Y., Rheological model of pore water, J. Rheology, 29, 131–145, (1985). Rice, C. L., and Whitehead, R., Electrokinetic flow in a narrow cylindrical capillary, J. Phys. Chem., 69, 4017–4024, (1965). Sasidhar, V., and Ruckenstein, E., Electrolyte osmosis through capillaries, J. Colloid Interface Sci., 82, 439–457, (1981). Sasidhar, V., and Ruckenstein, E., Anomalous effects during electrolyte osmosis across charged porous membranes, J. Colloid Interface Sci., 85, 332–362, (1982). Saville, D. A., Electrokinetic phenomena and anomalous conduction, Croat. Chem. Acta, 74, 1039–1047, (1998). Shaw, D. J., Introduction to Colloid and Surface Chemistry, 3rd ed., Butterworths. London, (1980). Stein, D., Kruithof, M., and Dekker, C., Surface-charge governed ion transport in nanofluidic channels, Phys. Rev. Lett., 93, 035901, (2004).

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Tang, T., Badal, M. Y., Ocvirk, G., Lee, W. E., Bader, D. E., Bekkaoui, F., and Harrison, D. J., Integrated microfluidic electrophoresis system for analysis of genetic materials using signal amplification methods, Anal. Chem., 74, 725–733, (2002). Taylor, G., Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. Roy. Soc. Lond., 219A, 186–203, (1953). Yang, J., Bhattacharyya, A., Masliyah, J. H., and Kwok, D. Y., Oscillating laminar electrokinetic flow in infinitely extended rectangular microchannels, J. Colloid Interface Sci., 261, 21–31, (2003). Yang, J., and Kwok, D. Y., Effect of liquid slip in electrokinetic parallel plate microchannel flow, J. Colloid Interface Sci., 261, 225–233, (2003). Yang, J., Surface effects of microchannel wall on microfluidics, Ph.D. Thesis, Department of Mechanical Engineering, University of Alberta, (2004). Yang, R. J., Fu, L. M., and Hwang, C. C., Electroosmotic entry flow in a microchannel, J. Colloid Interface Sci., 244, 173–179, (2001). Zholkovskij, E. K., Masliyah, J. H., and Czarnecki, J., Electroosmotic dispersion in microchannels with a thin double layer, Anal. Chem., 75, 901–909, (2003). Zholkovskij, E. K., and Masliyah, J. H., Hydrodynamic dispersion due to combined pressuredriven and electroosmotic flow through microchannels with a thin double layer, Anal. Chem., 76, 2708–2718, (2004). Zukoski, C. F., and Saville, D. A., The interpretation of electrokinetic measurements using a dynamic-model of the stern layer. I. the dynamic-model, J. Colloid Interface Sci., 114, 32–44, (1986).

CHAPTER 9

ELECTROPHORESIS

9.1 INTRODUCTION In the previous chapter, we discussed the flow behavior of an electrolyte past a stationary charged surface in presence of an external electric field. A similar, albeit, somewhat more complicated scenario arises when a colloidal particle bearing a surface charge immersed in an electrolyte solution is subjected to an external electric field. In this case, the particle is also susceptible to move in response to the imposed electric field, and a relative motion between the particle and the electrolyte solution is developed. Broadly, when an electric field is applied to a suspension of charged particles (or a single particle), the particles migrate along the field owing to the presence of surface charge on the particles. This phenomenon is referred to as electrophoresis. Electrophoresis is one of the most avidly studied electrokinetic transport phenomena, and provides a highly practical basis for separation of a mixture of charged colloidal particles and macromolecules, such as proteins, employing an electrical field. The literature on electrophoresis is rich in seminal contributions from numerous researchers dealing with the field of electrokinetic phenomena. These studies range from the earliest works of Smoluchowski (1918) and Henry (1931), where the attention was devoted to the motion of a single spherical colloidal particle under certain limiting conditions, to modern theories detailing the motion of a swarm of particles in a concentrated dispersion employing cell models (Levine and Neale, 1974a,b; Saville, 1977; Shilov et al., 1981; Kozak and Davis, 1986; Kozak and Davis, 1989a,b; Ohshima, 1997a; Dukhin et al., 1999; Carrique et al., 2001). In this chapter, we will Electrokinetic and Colloid Transport Phenomena, by Jacob H. Masliyah and Subir Bhattacharjee Copyright © 2006 John Wiley & Sons, Inc.

295

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briefly track the development of the theory of electrophoresis over the past century, primarily emphasizing the approaches that yield tractable analytical results. A brief overview of the numerical procedures will be provided in Chapter 14. The analysis to be presented is aimed at evaluation the electrophoretic velocity of a charged particle when it is placed in an electric field. First, we will consider the motion of a single charged spherical particle under the influence of an external electric field. After presenting the general mathematical construct describing such a situation, two limiting cases of the problem will be presented, namely, when the particle is very small or in an almost perfect dielectric with few mobile charges, i.e., in the limit κa ≪ 1, and when the particle is large or in a highly concentrated electrolyte, i.e., κa ≫ 1. We will then bridge these two limiting cases with a detailed analysis for intermediate values of κa (Henry, 1931). All the above mentioned analyses will be confined to particles having low surface potentials. A brief overview of perturbation approaches employed to predict the electrophoresis of highly charged particles will be presented. Following the analysis for single particles, we will present methodologies for addressing higher particle concentrations, representing dilute or highly concentrated colloidal dispersions. In the entire analysis, the particle will be assumed to be rigid and electrically non-conducting.1

9.2 ELECTROPHORESIS OF A SINGLE CHARGED SPHERE In this section, we will first provide a general description of the different types of transport phenomena encountered during electrophoresis. Following this, the electrophoretic mobility of a charged particle will be derived for two limiting cases. First, we will consider the electrophoresis of a single charged particle in the limit of κa ≪ 1, when the particle is small or when the electrolyte solution is infinitely dilute, containing vanishingly few mobile ions (free charges). Following this, we will consider the limit of κa ≫ 1, when the particle is large or the electrolyte solution is highly concentrated. 9.2.1 Transport Mechanisms in Electrophoresis A qualitative understanding of the general transport mechanisms encountered during electrophoresis is an important prerequisite for developing a mathematical framework to correctly predict the electrophoretic mobility. To develop this understanding, let us first consider a spherical particle of radius a bearing a charge Qs suspended in a pure dielectric fluid (containing no free charge or ions). When subjected to a uniform external electric field, E∞ , the particle will translate under the influence of the electric force acting on it. Since there is no free charge in the dielectric fluid, there will be no flow of this fluid under the influence of the uniform external electric field as long 1 Note that even a conducting particle will be rapidly polarized in the external electric field and will have no inner field. Consequently, such particles will also prevent passage of current through them, and appear as non-conducting (Probstein, 2003).

9.2 ELECTROPHORESIS OF A SINGLE CHARGED SPHERE

297

as there is no pressure gradient. The net electrical force on the charged particle will simply be FE = Qs E∞

(9.1)

As soon as the particle starts to move under the influence of this electrical force, it encounters an oppositely directed fluid drag force given by FH = 6π µaU

(9.2)

where µ is the fluid viscosity, and U is the particle velocity. Equating the electrical and drag forces for steady-state translation, one obtains U=

Qs E∞ 6π µa

(9.3)

The direction of the particle velocity will be governed by the sign of the particle charge. The electrophoretic mobility of the particle can be defined as velocity per unit applied electric field, and is given by η=

U Qs = E∞ 6π µa

(9.4)

Consider now an electroneutral electrolyte solution instead of a pure dielectric fluid subjected to a uniform external electric field, E∞ . First, consider the case when the charged particle is not immersed in this fluid. In this case, just as noted for the pure dielectric fluid, since the volumetric charge density, ρf , is zero everywhere in the electrolyte solution, there will be no electrical body force acting on the fluid. Consequently, there will be no flow of the electrolyte solution. However, there will be a migration or conduction current through the fluid that will obey Ohm’s law for ionic conductors. Let us next immerse the spherical particle of radius a carrying a total charge Qs in this electrolyte solution. The charged particle will polarize the electrolyte solution surrounding it, resulting in the formation of an electric double layer. The electric double layer will have a spatially varying volumetric charge density. As soon as a charge density is developed in the electrolyte solution, it will experience an electric body force under the influence of the overall electric field near the particle. The overall electric field is a superimposition of two fields: (i) the (generally unidirectional) external field and (ii) a spherically symmetric field in the double layer due to the charged particle in absence of the external field. The electric body force will cause a motion in the fluid immediately surrounding the particle. Noting that the net accumulation of the charge in the electric double layer will be opposite in sign to that of the particle charge, it is evident that the fluid flow will be opposite to the direction of the particle movement. In addition to the above mode of fluid flow, the concentration gradients of the ions, as well as the electric potential gradients will give rise to ionic fluxes, which are usually described in terms of the Nernst–Planck equations.

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ELECTROPHORESIS

It is therefore evident that a complete analysis of electrophoresis of a charged particle in an electrolyte solution involves consideration of three coupled physical processes. These are: (i) Interaction of the charged particle with the external electric field giving rise to an electrical force on the particle. (ii) Formation of the electric double layer surrounding the particle giving rise to a spatially inhomogeneous charge distribution, which results in an electrical body force driven fluid flow, and, (iii) Transport of ions relative to the charged particle under the combined influence of convection, diffusion, and migration. 9.2.2

General Governing Equations

The governing equations describing the three coupled processes mentioned above are presented here. The steady state equations presented here apply to the steady state velocity of a single charged particle under an imposed electric field, E∞ . The geometry is depicted in Figure 9.1. Note that the particle velocity, U, and the electric field are represented as vector quantities. To facilitate the problem solution, we translate the problem to a particle fixed reference frame, and consider the flow of an

Figure 9.1. Geometrical details required for writing the generalized governing equations for the electrophoretic mobility of a spherical particle of radius a held stationary in a uniform electric field E∞ .

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299

electrolyte solution around a stationary charged particle of radius a. In this case, far from the particle, the velocity of the electrolyte solution is given by u∞ = −U ix , where U is the particle velocity along the x direction and ix is a unit vector along the x coordinate. The velocity U represents the electrophoretic particle velocity when the particle is allowed to move in a quiescent fluid. Consequently, u∞ = −U represents the fluid velocity necessary to keep the particle stationary. The electrolyte solution is a Newtonian fluid having a viscosity µ. The electrolyte solution contains N ionic species of valency zi with a bulk concentration ni∞ and diffusion coefficient Di . The electric potential, ψ, in the electrolyte solution surrounding the charged particle and the space ionic charge density, ρf , in the electrolyte solution are related by Poisson’s equation, (5.7), as ǫ∇ 2 ψ = −ρf

(9.5)

where the space charge density [Eq. (5.9)] is given by ρf =

N


(9.6)

zi eni

i=1

For small Reynolds number and neglecting the effect of gravity, the electrolyte solution velocity field, u, is given by the modified momentum equation, as was presented by Eq. (6.112) µ∇ 2 u − ∇p − ρf ∇ψ = 0 (9.7) where ∇p is the pressure gradient and ρf ∇ψ is the electrical body force on the electrolyte solution caused by the overall potential gradient, ∇ψ. The steady-state continuity equation for the electrolyte solution under dilute conditions, i.e., constant density, Eq. (6.113), provides ∇ ·u=0

(9.8)

The ionic fluxes are given by the Nernst–Planck equations, which can be written in terms of the number concentration, ni , of the i th ionic species as [cf., Eq. (6.45)] j∗∗ i = ni vi = ni u − Di ∇ni −

zi eni Di ∇ψ kB T

(9.9)

where vi and Di are the velocity and diffusivity of the i th ionic species, respectively. For a steady state transport process, the ion conservation equation then yields   zi eni Di ∗∗ ∇ · ji = ∇ · ni u − Di ∇ni − (9.10) ∇ψ = 0 kB T 9.2.3

Boundary Conditions

The shear plane of the electric double layer is assumed to coincide with the surface of the sphere. At the sphere surface, the electrical boundary condition for the Poisson

300

ELECTROPHORESIS

equation can be defined as (9.11)

−ǫ n · ∇ψ = qs

where qs is the surface charge density of the particle. Typically, for an isolated spherical particle, one can relate the surface charge density to the surface potential using independent expressions such as Eq. (5.83). Assuming the spherical particle to be stationary, the no slip condition for the Navier–Stokes equations becomes u=0

(9.12)

Finally, for the Nernst–Planck equations, we impose the condition that no electrolyte ions penetrate the sphere surface   zi eni Di vi · n = Di ∇ni + ∇ψ · n = 0 kB T

and

u·n =0

(9.13)

In the above equations, n is a unit outward normal vector on the sphere’s surface. Far away from the sphere, the following boundary conditions hold. The electrical boundary condition is ∇ψ = −E∞

(9.14)

For the Navier–Stokes equations, the appropriate far field boundary condition is that the overall fluid stress (both pressure and viscous stresses) vanishes. Accordingly, the total fluid stress at every point of the far field boundary is expressed using    = = σ · n = −p I + µ ∇u + (∇u)T · n = 0

(9.15)

ni = ni∞

(9.16)

where p is the hydrostatic pressure. Finally, for the Nernst–Planck equations, the ion concentrations in the far field are given by their bulk values

 which should satisfy the bulk electroneutrality condition, zi eni∞ = 0. Note that the formulation of the problem does not specify the far field electrolyte solution velocity. The fluid velocity far away from the particle, u∞ , is determined from the solution of the governing equations in the particle fixed coordinate system. From this far field fluid velocity, the particle velocity, U , can be deduced noting that u∞ = −U ix

(9.17)

For an N component electrolyte system, a self-contained solution of the general governing equations will consist of the fluid velocity vector u, the pressure, the electric potential ψ, and the ionic concentration, ni , for all the N species. There are thus N + 3 unknowns, assuming the velocity vector to be a single quantity, and that the surface charge density of the particle, qs , is an independently known constant. Equations (9.7)

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301

and (9.8) provide the equations corresponding to the velocity vector and pressure, Eq. (9.10) provides an equation for each of the ion concentrations (there will be N such equations) while Eqs. (9.5) and (9.6) provide the potential distribution, ψ. The overall solution of these equations (for steady-state flow) will self-consistently satisfy the force balance on the particle, which involves equalizing the electrical (Maxwell) and hydrodynamic forces acting on the particle. Unfortunately, a complete solution of the above general equations is only possible through numerical techniques. We will discuss such a numerical solution methodology in Chapter 14. Approximate analytical solutions of the above transport model are generally based on perturbation approaches and involve several approximations. When considering the general problem formulation described above, it should be noted that the formulation is only based on the “outer problem” pertaining to the electrolyte solution surrounding the particle. This forced us to assume the surface charge density (or the surface potential) to remain unaltered in the presence of the external electric field. Such a simplification does not lead to difficulties as long as the externally imposed electric field is small compared to the electric field surrounding the particle due to the electric double layer. However, the formulation as stated above becomes inadequate when considering large external fields. For a rigorous modelling of the problem, one additionally needs to obtain the electrical potential distribution and fields inside the particle. This is usually derived from the Laplace equation written within the particle when there is no free charge inside the particle. The electric boundary conditions at the particle-fluid interface then evolve from the conditions of continuity of the potential and the discontinuity of electric displacement as discussed in Chapters 3 and 4. The simplification of the governing equations to render them amenable to analytical solution techniques requires a qualitative understanding of some key features of the electrophoretic transport. The simplifications of the model equations should account for these transport mechanisms, otherwise the electrophoretic mobility predictions will be of limited accuracy. For a finite-thickness electric double layer around a charged particle, three effects need to be considered (Shaw, 1980). These are: Electrophoretic Retardation: A charged particle will polarize the electrolyte solution in its immediate vicinity giving rise to a countercharged ion cloud. When this charge cloud with an effective volumetric charge density, ρf , is subjected to an externally imposed electric field, E∞ , it will create a body force in the fluid of magnitude ρf E∞ . The fluid surrounding the particle will move under the influence of this body force. Since the ion cloud will be oppositely charged compared to the particle, the overall fluid motion will be opposite to the direction of the particle motion. The oppositely directed motion of the fluid tends to retard the movement of the particle, hence the term electrophoretic retardation. Solving the Navier–Stokes equation together with the electric force term would account for this effect. Relaxation: The charge cloud surrounding the particle in presence of the external electric field is not symmetric as in the case of an equilibrium electric double layer around a stationary particle. Since, in general, the particle has a different dielectric permittivity and conductivity than the surrounding fluid, the external electric field

302

ELECTROPHORESIS

will give rise to a polarization charge (induced charge) and an associated field at the particle surface. Such an induced charge relaxes through diffusion and migration mechanisms. Furthermore, due to the movement of the particle under consideration relative to the mobile ions in the double layer, the distribution of the ions around the particle is not symmetric. A finite time (relaxation time) is required for the original symmetry to be restored by convection, diffusion, and migration of the ions. This asymmetry gives rise to a retardation force called the relaxation effect. If one takes into account the Nernst–Planck equation in the analysis, i.e., non-zero Peclet numbers, the relaxation effects would be included. When κa < 0.1, the relaxation effects are not important. For large zeζ /kB T , the relaxation effect is important. The shape of the electric double layer with the relaxation effect is schematically shown in Figure 9.2. Surface Conductance: Due to the presence of a charged surface, the distribution of the ions in the mobile region of the electric double layer gives rise to a higher conductivity close to the surface than that in the bulk electrolyte. When ζ and κa are small, the electric conductivity in the double layer is close to that of the bulk electrolyte. When κa or ζ is not small, the electrophoretic mobility of a particle is affected by the different electric conductivity in the double layer. This effect is called the surface conductance. If the surface conductance is important (large κa and/or ζ ), the calculated zeta potentials may be quite low (Shaw, 1980). The general governing equations account for all the transport effects mentioned above. However, approximate solutions often neglect one or more of the above mentioned transport mechanisms. Typically, all approximate solutions start from the assumption that the external field is much smaller in magnitude compared to the field produced by the charged particle. Thus, all the approximate analytical solutions

Figure 9.2. Double layer distortion due to relaxation effect. Translation of a negatively charged particle.

9.2 ELECTROPHORESIS OF A SINGLE CHARGED SPHERE

303

consider small perturbations in the equilibrium double layer surrounding the charged particle due to the external field. The manner in which the external field is considered to perturb the equilibrium electric double layer results in different analytic results of different range of applicability. In the following, we will first consider two of the simplest approaches that provide some limiting perspective of the electrophoretic particle mobility for the limiting cases of small and large κa. 9.2.4

Electrophoresis in the Limit κa ≪ 1

Consider a spherical non-conducting colloidal sphere of radius a placed in an electrolyte solution subject to an external electrical field E∞ as shown in Figure 9.3. Although we consider presence of free charge in the form of mobile ions, the concentration of these ions are sufficiently low to make the Debye screening length κ −1 very large compared to the particle radius. In this limit, κa ≪ 1, the expression for the electrophoretic mobility of the particle will be derived. Figure 9.3 shows the geometry under consideration. Using a spherical coordinate system, and utilizing symmetry in the angular directions, the potential distribution around the sphere can be represented by the Poisson–Boltzmann equation. Assuming a symmetric (z : z) electrolyte, the governing Poisson–Boltzmann equation, (5.68), in the Debye–Hückel limit becomes 1 d r 2 dr



dψ r2 = κ 2ψ dr

(9.18)

where κ is the inverse Debye length, given by κ=



2e2 z2 n∞ ǫkB T

1/2

Figure 9.3. A schematic representation of electrophoresis of a spherical particle for κa ≪ 1.

304

ELECTROPHORESIS

Let ξ = rψ

(9.19)

d 2ξ = κ 2ξ dr 2

(9.20)

which renders Eq. (9.18) as

The general solution of the above equation is ξ = Ae−κr + Beκr

(9.21)

where A and B are constants, which can be evaluated from the boundary conditions. The boundary conditions are ψ → ψ∞ = 0

as r → ∞

(9.22)

at r = a

(9.23)

and ψ → ψs = ζ

Note the use of the zeta potential, ζ , at the particle surface (r = a) instead of the actual surface potential ψs of the particle in Eq. (9.23). Substituting the boundary conditions, Eqs. (9.22) and (9.23) in Eq. (9.21) yields the potential distribution as ψ =ζ

a exp[−κ(r − a)], r

(9.24)

where a is the particle radius. Use of the Poisson–Boltzmann equation to obtain the potential distribution implies that the electrolyte solution is stationary. It should be recalled that the assumption of a zero Peclet number was used to obtain the Boltzmann distribution and hence Eq. (9.18). This assumption implies that the electrolyte velocity does not affect the ionic equilibrium. We are now in a position to determine the charge on the particle surface, Qs . In Chapter 5, we noted that the total charge of a system comprised of a charged particle and the surrounding electrolyte should be zero (owing to electroneutrality). Using this information, the surface charge, Qs , of a spherical particle of radius a is given by Qs = −



∞

(4π r 2 )ρf dr

(9.25)

a

where the right hand side of Eq. (9.25) denotes the total amount of free charge in the electrolyte solution. Using Poisson’s equation, 1 d r 2 dr



r

2 dψ

dr



=−

ρf ǫ

(9.26)

305

9.2 ELECTROPHORESIS OF A SINGLE CHARGED SPHERE

Eq. (9.25) becomes Qs = 4π ǫ



a

∞

r

2



1 d r 2 dr



r

2 dψ

dr





dr = 4π ǫ r

2 dψ

dr

∞

(9.27)

a

Noting that dψ →0 dr

at r → ∞, we obtain

dψ Qs = −4π a ǫ dr r=a 2

(9.28)

The term dψ/dr can be readily obtained by taking the radial derivative of ψ with respect to r, Eq. (9.24). Using this derivative in Eq. (9.28) gives Qs = 4π a 2 ǫζ

(1 + κa) a

(9.29)

The electrical force acting on the particle in an external field E∞ is given by FE = Qs E∞ = 4π a 2 ǫζ E∞

(1 + κa) = 4π aǫζ E∞ (1 + κa) a

(9.30)

As soon as the sphere starts to translate due to the electrical force, an electrolyte solution induced drag force counteracts this motion. However, to consider the drag force acting on the particle, one needs to first note that the charged fluid in the electric double layer surrounding the particle will acquire an oppositely directed velocity under the influence of the external electric field. The velocity of the center of the spherically symmetric charge cloud can be computed using the singular solution of Stokes equation discussed earlier in Section 6.3. Recall that the Oseen formulation is used to calculate the velocity field due to a point force acting at a given location in the fluid, and that the overall velocity field due to distributed point forces is obtained by integrating the singular solutions of the Stokes equation. Here, the point force is given as the product of the external field, E∞ , and the differential charge, dQ = ρf dV , of a small volume element dV in the fluid. From this analysis, it can be shown that the velocity component acting along the field direction is given by   E ∞ ρf  UR = 1 + cos2 θ dV (9.31) V 8π µr where UR is the velocity of the center of the spherically symmetric charge cloud, V is the volume of the charge cloud, r is the radial position, and θ is the polar angle. Performing the integration over the entire volume of the charge cloud, one obtains the velocity of the center of the charge cloud as UR = −

2 ǫζ (κa)E∞ 3 µ

(9.32)

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ELECTROPHORESIS

This is the retardation velocity of the fluid surrounding the particle, and it approaches zero as κa → 0. In an infinitely dilute colloidal dispersion (where the particle volume fraction, φp → 0), the Stokes hydrodynamic drag force on the spherical particle of radius a is given by FH = 6π µa(U − UR )

(9.33)

where U − UR is the velocity of the particle relative to the electrolyte solution and µ is the viscosity of the electrolyte solution. For a non-accelerating particle, balancing the electrical and drag forces, i.e., FE = FH , results in U=

2 ǫζ E∞ 3 µ

(9.34)

This is the Hückel (1924) solution for the electrophoretic velocity valid for small κa. Equation (9.34) is usually written as η=

2 ǫζ U = E∞ 3 µ

(9.35)

where η = U/E∞ is the velocity per unit field strength, and is referred to as the electrophoretic mobility. Its unit is (m2 V−1 s−1 ) or (C s kg−1 ). As would be expected, increasing the surface potential ζ or decreasing the electrolyte solution viscosity µ would increase the electrophoretic mobility. The Hückel equation is valid for large Debye length and, in particular, for non-electrolyte systems, e.g., organic liquids, where κ −1 is generally large relative to the particle radius, a. The Hückel solution accounts for the retardation effect, but assumes a spherically symmetric double layer, and hence, does not consider any relaxation effect. 9.2.5

Electrophoresis in the Limit κa ≫ 1

When the particle radius is large, or more specifically, when the Debye screening length κ −1 is small relative to the particle radius, i.e., κa ≫ 1, the electric double layer becomes extremely thin compared to the particle radius, and we can neglect the curvature effects of the particle. In this limiting case, one can consider the relative movement of the ions with respect to a planar surface. In other words, the electrophoretic problem reverts to the case of an electrolyte flowing past a planar surface with the externally imposed field aligned parallel to the surface as shown in Figure 9.4. The analysis based on this assumption is generally referred to as the Helmholtz–Smoluchowski analysis. Consider the governing Navier–Stokes equation for the fluid flow past a horizontal flat surface, with the surface moving at a velocity U . Referring to Figure 9.4(b), we consider a coordinate system fixed on the particle surface. In this particle fixed coordinate frame, the fluid velocity at the particle surface is assumed to be zero, and the fluid velocity far away from the particle surface is given by −U . Based on the

9.2 ELECTROPHORESIS OF A SINGLE CHARGED SPHERE

307

Figure 9.4. A schematic representation of electrophoresis of a positively charged spherical particle for κa ≫ 1. In this case, the relative motion between the electrolyte and the particle shown in (a) can be recast as the electroosmotic transport problem of the flow of the electrolyte past a stationary planar surface with the electric field aligned parallel to the surface (b). The direction of the fluid flow in (b) is opposite to the direction of the particle motion in (a).

geometry of Figure 9.4 and Table 6.7, we obtain µ

d 2 ux = −ρf E∞ dy 2

(9.36)

Here ux is the velocity of the fluid tangential to the plate, ρf is the charge density in the fluid, and E∞ is the externally imposed electric field. Combining with the Poisson equation, ǫ

d 2ψ = −ρf dy 2

(9.37)

308

ELECTROPHORESIS

Eq. (9.36) becomes µ

d 2ψ d 2 ux = ǫE ∞ dy 2 dy 2

(9.38)

The boundary conditions for the problem are as follows: ψ →0 dψ →0 dy ψ =ζ

and

ux → −U dux and →0 dy and ux = 0

as y → ∞

(9.39-a)

as y → ∞

(9.39-b)

at y = 0

(9.39-c)

Integrating Eq. (9.38) from infinity to an arbitrary distance y from the particle surface, and employing the boundary conditions (9.39-a) and (9.39-b), yields ux (y) =

ǫE∞ ψ(y) −U µ

(9.40)

Substituting the boundary condition at the particle surface, Eq. (9.39-c), in Eq. (9.40) results in ǫζ E∞ U= µ or, expressed in terms of the electrophoretic mobility, η=

U ǫζ = E∞ µ

(9.41)

Equation (9.41) is referred to as the Helmholtz–Smoluchowski electrophoretic mobility equation, and is valid in the limit κa ≫ 1 and zeζ /kB T < 1. This equation is also referred to as Smoluchowski’s electrophoretic mobility equation in the literature. It should be noted that the Hückel and the Helmholtz–Smoluchowski results, given by Eqs. (9.34) and (9.41), respectively, provide the two limiting values of the electrophoretic velocity of a single colloidal particle as κa → 0 and κa → ∞, respectively. These two expressions differ by a factor of 2/3. Note that both the Hückel and Helmholtz–Smoluchowski (H–S) equations are independent of particle size. Moreover, since the H–S equation (for large κa) was derived by assuming a flat surface, i.e., no curvature effect, we can conclude that it should also be valid for any shape provided that the electric double layer is very thin everywhere and that the particle is non-conducting.

9.3 IMPROVED SOLUTIONS: ARBITRARY DEBYE LENGTH The electrophoretic velocities derived so far are for either κa ≪ 1 or κa ≫ 1. In particular, the H–S equation was derived with the particle surface being treated as

9.3 IMPROVED SOLUTIONS: ARBITRARY DEBYE LENGTH

309

locally flat. On the other hand, in the derivation of the Hückel equation, any perturbation of the electric double layer due to the influence of the external electrical field (the relaxation effect) was neglected. While neglecting the curvature of the particle and the ionic movement effects in the two limiting solutions is quite reasonable, these effects cannot be ignored when considering the electrophoretic velocities of a charged particle at intermediate values of the parameter κa. Numerous solutions of the electrophoretic velocity of a charged particle for intermediate values of κa have been developed. In the following, we will discuss the solution methodologies accounting for the electrophoretic retardation and relaxation effects employing the linear perturbation approach. First, the general governing equations pertaining to the perturbation technique are presented. We then briefly describe the approach due to Henry (1931), which allows calculation of the electrophoretic mobility for the entire range of κa (0 < κa < ∞). The approach of Henry (1931) is however limited to low surface potentials, and ignores the relaxation effect arising from ionic convection. A brief overview of the perturbation approaches accounting for the relaxation effects and higher surface potentials on the particle is also presented in this section (Overbeek, 1943; Booth, 1950; Wiersema et al., 1966; O’Brien and White, 1978). 9.3.1

Perturbation Approach

The coupled non-linear governing equations outlined in Section 9.2.1 pose a formidable challenge in terms of obtaining a direct analytic solution. For the cases where the applied electrical field E∞ is small compared to the electrical fields inside the double layer, one can assume that the electrical double layer is only slightly distorted from the equilibrium configuration due to the applied field and the ensuing particle motion. In this case, the governing equations, Eqs. (9.5)–(9.10), as well as their associated boundary conditions, can be replaced by their approximate linearized forms using a perturbation approach. To derive these linearized equations, the unknown variables in the governing equations are represented as a sum of the equilibrium value in absence of the external field (see Section 6.2.7) and a perturbation due to the external field. The perturbation term is considered to be directly proportional to the external field. Thus, one can write ψ = ψ eq + δψ

(9.42-a)

u = 0 + δu

(9.42-b)

eq

+ δp

(9.42-c)

eq ni

+ δni

(9.42-d)

p=p ni =

Substituting the above variables in the general governing equations, (9.5)–(9.10), and retaining up to the linear terms, one obtains two sets of equations. First, retaining only the equilibrium terms, one obtains
eq ǫ∇ 2 ψ eq = −ρf = − zi eni (9.43)

310

ELECTROPHORESIS

for the Poisson equation, 0 = −∇p eq + ǫ∇ 2 ψ eq ∇ψ eq

(9.44)

for the modified Navier–Stokes equation, and ∇·



eq ∇ni

 eq zi eni eq ∇ψ =0 + kB T

(9.45)

for the Nernst–Planck equation. The boundary conditions for the equilibrium problem are set as follows: At the particle surface, Eq. (9.11) gives −ǫn · ∇ψ eq = qs

(9.46)

while Eq. (9.13) yields 

eq

∇ni +

 zi e eq ni ∇ψ eq · n = 0 kB T

(9.47)

Far away from the particle, r → ∞, we have eq

ni = ni∞

and ψ eq = 0

(9.48)

Solution of the equilibrium equations (9.43) to (9.45) subject to the above boundary conditions provides the Poisson–Boltzmann description of the electrostatic problem around the stationary particle in absence of any external electric field, and hence, any ionic or fluid movement. Solution of the above stationary problem was discussed in detail in Chapter 5. The equations containing the linear perturbation terms are written next. In writing these equations, only the linear perturbation terms are retained, and the products of the perturbation terms are neglected. The Poisson equation in terms of the perturbed variables becomes
ǫ∇ 2 δψ = − zi eδni (9.49)

The Navier–Stokes equation becomes
eq
zi ni ∇δψ − e zi δni ∇ψ eq = 0 µ∇ 2 δu − ∇δp − e

(9.50)

with

∇ · δu = 0

(9.51)

for the continuity equation. Finally, the perturbed form of the Nernst–Planck equation becomes   zi eDi eq eq eq (δni ∇ψ + ni ∇δψ) = 0 (9.52) ∇ · ni δu − Di ∇δni − kB T

9.3 IMPROVED SOLUTIONS: ARBITRARY DEBYE LENGTH

311

For this problem, the boundary conditions at the particle surface are δu = 0 

− n · ∇δψ = 0   zi e eq eq n ∇δψ + δni ∇ψ ·n =0 ∇δni + kB T i

(9.53-a) (9.53-b) (9.53-c)

Far from the particle, the boundary conditions at r → ∞ are ∇δψ = −E∞

(9.54-a)

δni = 0    = −δpI + µ ∇δu + (∇δu)T · n = 0

(9.54-b) (9.54-c)

Equations (9.49)–(9.52) are the linearized forms of the governing transport equations, and are traditionally the starting points for theoretical investigations employing perturbation methods in electrophoresis. Simplifications are made along the way, which usually involve expanding the perturbation variables as a Taylor series in terms of the scaled particle surface potential, s = zeζ /kB T . Henry (1931) derived an analytical expression for the electrophoresis of a single spherical particle that is valid for small zeta potentials (the Debye–Hückel approximation) and for an arbitrary double-layer thickness which considers the electrophoretic retardation effect. However, the relaxation effects arising from ionic convection was neglected. Overbeek (1943) and Booth (1950) were the first to correctly account for the ionic convection effect about a spherical particle but their results are only valid for small zeta potentials (zeζ /kB T ). Wiersema et al. (1966) relaxed the constraint of a small zeta potential and numerically solved the governing equations for the electrophoresis of a non-conducting sphere. Ionic convection was considered. Numerical difficulties at high values of ζ were encountered. Subsequently, O’Brien and White (1978) numerically solved the governing equations for arbitrary ζ and κa. Their work represents a proper solution of the electrophoresis problem for a sphere (Kozak and Davis, 1989a). 9.3.2

Henry’s Solution

As stated earlier, the limiting results for κa ≫ 1 and κa ≪ 1 given by the Helmholtz– Smoluchowski and the Debye–Hückel expressions, respectively, were bridged by Henry (1931) for intermediate values of the parameter κa. One can derive Henry’s result starting from the linear perturbation equations, (9.49)–(9.54-c). Such an approach requires expanding the perturbation variables discussed in Section 9.3.1 in terms of the scaled particle surface potential, and retaining the leading order terms. However, we will not discuss the perturbation approach here. Instead, we will simply reiterate the approach originally presented by Henry (1931), reserving some comments on its validity for subsequent discussion. Henry developed the

312

ELECTROPHORESIS

theory from a general set of fluid mechanical and electrostatic equations describing the electrophoretic transport problem. Two main assumptions were made in Henry’s approach: 1. The electric double layers are not distorted by the flow. The total potential within the double layer is a linear combination of the electric double layer potential and the potential due to the externally imposed electric field. 2. The Debye–Hückel approximation for low surface potential is assumed. Henry’s approach for solving the governing transport equations of Section 9.3.1 was to consider the equilibrium shape of the electrical double layer (no distortion of the ionic cloud due to the movement of the particle). Due to Henry’s assumptions, effects arising from convective relaxation and surface conductance are not accounted for in the analysis. In this case, the governing equations for ion transport, Eq. (9.10) simplify to provide the equilibrium Boltzmann distribution of ions. Consider a sphere of radius a, electrical conductivity σ ′ and surface potential ζ immersed in an infinite volume of a Newtonian electrolyte solution with an electrical conductivity σ ∞ , dielectric permeability ǫ, and viscosity µ. If an external field E∞ is applied to the system in a direction parallel to the x-axis, the particle will be subject to an electrophoretic velocity U along the positive x-direction. Henry considered the situation when the particle is immobile (i.e., stationary) and the electrolyte solution flows past it with a velocity U along the negative x-direction. The geometry of the problem is depicted in Figure 9.5. It is assumed that the problem can be decomposed into two parts: One part is to solve for a spherically-symmetric potential distribution around the charged sphere in an electrolyte solution containing free charges. The

Figure 9.5. Electrophoresis with no relaxation effects. A stationary particle with the solvent (electrolyte) flowing along the negative x direction past it with a velocity U .

9.3 IMPROVED SOLUTIONS: ARBITRARY DEBYE LENGTH

313

second part is to solve for a charged particle in a dielectric having no free charge but with an external electric field being applied. At the outset, we note that Henry developed his mathematical formulation assuming finite conductivities for the particle and the suspending medium. However, to facilitate comparison with the Smoluchowski and Hückel results, his final results were provided for a non-conducting sphere, σ ′ = 0. Accordingly, we will derive his results here assuming a non-conducting spherical particle. The potential due to the charged spherical particle is given by2 ǫ∇ 2 ψ = −ρf

(9.55)

Neglecting θ -variation, making use of the fact that zeζ /kB T ≪ 1 and assuming that the Boltzmann distribution is valid, Eq. (9.55) becomes

1 d 2 dψ (9.56) r = κ 2ψ r 2 dr dr where κ= The boundary conditions are



2e2 z2 n∞ ǫkB T

1/2

r →∞ ψ → 0 r= a

ψ =ζ

The solution is given by a 

(9.57) e−κ(r−a) r The potential given by Eq. (9.57) is solely due to the electric double layer. No applied electric field is being considered. Consider now a particle in a charge-free dielectric. The potential due to the external field is φ. The governing equation becomes ψ =ζ

∇ 2φ = 0

(9.58)

E = −∇φ

(9.59)

with Equation (9.58) can be written as (see Table 6.9)



∂ 1 ∂φ 1 ∂ 2 ∂φ r + sin θ =0 r 2 ∂r ∂r r 2 sin θ ∂θ ∂θ 2

(9.60)

In this section, we denote the potential due to the stationary electric double layer as ψ instead of ψ eq . We do this to simplify the notation used for the mathematical development.

314

ELECTROPHORESIS

The boundary conditions are BC-1: φ = −E∞ r cos θ for r → ∞ ∂φ = 0 at r = a BC-2: ∂r

(9.61) (9.62)

BC-2 is obtained by recognizing that the dielectric permittivity of the particle is insignificant compared to that of the surrounding fluid (which is generally the case). Also, at the particle surface, jir⋆⋆ = 0, ur = 0 and ni = 0. E∞ is the undisturbed electric field strength. Solution of this electrostatic problem was provided in great detail in Section 4.2. The reader is advised to refer to that solution at this point. The solution of Eq. (9.60) subject to the boundary conditions of Eqs. (9.61) and (9.62) is given by   a3 φ = −E∞ r + 2 cos θ (9.63) 2r It is now necessary to evaluate the hydrodynamic and electric forces exerted on the charged particle. The hydrodynamic forces can be evaluated from the solution of the Navier–Stokes equation modified for the electric force. The momentum equation for creeping flow is given by µ∇ 2 u − ∇p = ρf ∇(φ + ψ)

(9.64)

which needs to be solved together with the continuity equation ∇ ·u=0

(9.65)

The following boundary conditions apply for the fluid flow problem ur = −U cos θ uθ = U sin θ

as r → ∞

(9.66)

as r → ∞

(9.67)

at r = a

(9.68)

and ur = uθ = 0

As mentioned earlier, the spherical particle is held stationary with the fluid flowing at a velocity of U along the negative x direction. The potentials ψ and φ are given by, respectively, Eqs. (9.57) and (9.63). The solution to Eqs. (9.64) and (9.65) subject to the boundary conditions of Eqs. (9.66), (9.67) and (9.68) was obtained by Henry (1931). The mathematical manipulations are quite involved. The solution provides expressions for p, ur , and uθ as functions of r and θ. The hydrodynamic force FH exerted by the fluid on the particle can be written as FH = 2π a

2



0

π

[−τrr cos θ + τrθ sin θ]r=a sin θ dθ

(9.69)

9.3 IMPROVED SOLUTIONS: ARBITRARY DEBYE LENGTH

315

where τrr = −p + 2µ

∂ur ∂r

(9.70)

and  ∂uθ uθ 1 ∂ur τrθ = µ − + (9.71) ∂r r r ∂θ In Eq. (9.69) τrr and τrθ represent the normal and tangential stresses exerted by the fluid on the particle. Figure 9.6 shows the hydrodynamic surface forces. Note that FH is taken as positive when measured along the velocity U (in opposite direction to x). The electric force due to the electric field is obtained by integrating the Maxwell stress tensor over the particle surface. This force can also be represented as the product of the total particle charge and the electric field strength E∞ . Let us derive the electrical force term. Consider a charged spherical particle of radius a located in a dielectric containing free charge; the electric force in the x-direction, FE , exerted on the particle is given by  FE = [qs Ex ]a dS (9.72) 

S

where qs is the surface charge density (charge per unit surface area, C/m2 ). The subscript x denotes x-direction and subscript a is for r = a. Let us first determine qs and Ex on the particle surface. For the special case of no angular variation in ψ, the charge density qs follows from Eq. (9.28), and is written as dψ Qs = −ǫ (9.73) qs = 4π a 2 dr r=a

Figure 9.6. Hydrodynamic and electric surface forces on a particle.

316

ELECTROPHORESIS

The electric field strength Er is given by Er = −

∂ (ψ(r) + φ(r, θ)) ∂r

(9.74)

Since at r = a, ∂φ/∂r = 0 [Eq. (9.62)], we have at the surface of the sphere Er = −

dψ dr

(independent of θ)

(9.75)

Now Eθ = −

1 ∂φ 1 ∂ (ψ + φ) = − r ∂θ r ∂θ

(9.76)

The term ∂ψ/∂θ = 0 because of the assumption that there is no θ-dependence for ψ. The electric field strength in x-direction is given by Ex = −Eθ sin θ + Er cos θ

(9.77)

Making use of Eqs. (9.75) and (9.76), we obtain Ex =

1 ∂φ dψ sin θ − cos θ r ∂θ dr

(9.78)

Making use of Eqs. (9.73) and (9.78), the electric force in x-direction as provided by Eq. (9.72) is given by FE =



0

π



dψ − ǫ dr



 dψ 1 ∂φ sin θ − cos θ [2π a sin θ] a dθ a ∂θ dr

(9.79)

As the integral 

π

cos θ sin θ dθ = 0

0

The term (dψ/dr) cos θ of Eq. (9.79) does not contribute to the force term, and consequently, Eq. (9.79) reduces to FE = −2π aǫ

 π dψ ∂φ sin2 θ dθ dr a 0 ∂θ a

(9.80)

9.3 IMPROVED SOLUTIONS: ARBITRARY DEBYE LENGTH

317

From Eq. (9.63), we obtain

and Eq. (9.80) becomes

∂φ 3a E∞ sin θ = ∂θ a 2 2

FE = −3π a ǫE∞

 π dψ sin3 θ dθ dr a 0

(9.81)

The electric force on the particle in x-direction is then given by  dψ E∞ FE = −4π a ǫ dr r=a 

2

(9.82)

Equation (9.82) gives the electric force in Henry’s solution. Here the term in brackets is the total surface charge. Note that φ does not contribute to the force FE , as it acts in a charge free dielectric. At steady state, the sum of the hydrodynamic and electric forces is zero as the particle does not have any acceleration. This leads to FH = FE

(9.83)

 2ǫζ E∞ f (κa) 3µ

(9.84)

and U=



where 1 5 1 1 (κa)2 − (κa)3 − (κa)4 + (κa)5 16 48 96 96

 ∞ −t

1 e (κa)2 4 κa + (κa) e 1− dt 8 12 t κa

f (κa) = 1 +

(9.85)

Equation (9.84) can be rearranged to express the electrophoretic mobility as η=

  2ǫζ U = f (κa) E∞ 3µ

(9.86)

Note that the integral in the last term of Eq. (9.85) is an exponential integral, and a simple numerical quadrature formula (like trapezoidal or Simpson’s rules) is generally not sufficient to evaluate it. One should either employ appropriate numerical integration schemes specially designed for evaluating exponential integrals, or should use asymptotic series expansions (which are available for large values of κa). A plot of the function f , usually called Henry’s function, is given in Figure 9.7.

318

ELECTROPHORESIS

Figure 9.7. Variation of Henry’s function with κa. Solid line, Eq. (9.85); dashed line, Eq. (9.87).

From Figure 9.7 it can be seen that for κa ≪ 1, f (κa) → 1, and the electrophoretic velocity given by Eq. (9.84) reduces to Hückel’s expression, Eq. (9.35). For the case of κa ≫ 1, i.e., a thin electric double layer f (κa) → 3/2 and Eq. (9.84) gives the electrophoretic velocity as given by Eq. (9.41) for the Helmholtz–Smoluchowski expression. Strictly speaking, Henry’s equation is valid for small values of surface potential: i.e., zeζ /kB T ≪ 1 for all κa values. It should also be noted that in Henry’s derivation, it was assumed that the fluid moves with a velocity U while the particle is held stationary in the electric field. Table 9.1 gives values for Henry’s function as calculated using Eq. (9.85). A curve fit equation for f (κa) is given by f (κa) =

3 1 − 2 2[1 + a1 (κa)a2 ]

(9.87)

where a1 = 0.072 and a2 = 1.13. It is perhaps pertinent at this point to identify certain conceptual inconsistencies with Henry’s approach described above. It is conceptually incorrect to assume that the potential distribution engendered by the external field obeys the Laplace equation as discussed in this section leading to the potential distribution given by Eq. (9.63). It is also incorrect to assume that the external potential, φ, can be linearly superimposed on the potential due to the charged particle, ψ. In fact, the perturbation equations described in Section 9.3.1 do not lead to such a linear superposition of the electrical potentials. Instead, after representing the ionic concentrations in terms of

9.3 IMPROVED SOLUTIONS: ARBITRARY DEBYE LENGTH

319

TABLE 9.1. Values for Henry’s Function Evaluated Numerically Using Eq. (9.85). κa

f (κa)

0 0.5 1.0 2.0 3.0 4.0 5.0 10.0 25.0 50.0 100.0 ∞

1.000 1.009 1.027 1.065 1.101 1.132 1.160 1.253 1.365 1.423 1.458 1.500

the electrochemical potentials through µi = µi0 + kB T ln ni + zi eψ

(9.88)

where µi is the i th ion electrochemical potential, one can derive a perturbation variable of the form kB T δni δµi = + δψ (9.89) δφi = zi e zi e neq i It is this perturbation variable, δφi , which eventually leads to a Laplace equation ∇ 2 δφi = 0

(9.90)

when substituted in the perturbed form of the Nernst–Planck equation. Such a formulation leads to a solution for δφi that is identical in form to Eq. (9.63). The subsequent solution of the perturbation equations is quite similar to the approach originally developed by Henry, leading to the same result for the electrophoretic mobility as given by Eq. (9.86). Thus, in retrospect, although Henry’s formulation might be construed as conceptually flawed, the analysis leads to a correct final result for the electrophoretic mobility. Indeed, if one performs the linear perturbation analysis, it becomes evident that Henry’s solution accounts for the diffusion-migration based relaxation effects (it only excludes the convective relaxation effect). 9.3.3

Effect of Particle Conductivity and Shape

The results of Henry (1931) discussed above dealt with a non-conducting sphere. The conductivity of a particle affects (distorts) the applied electric field. In this context, one should first note that the subsequent discussion refers to particles in which the

320

ELECTROPHORESIS

conductivity arises due to the presence of free charge carriers (ions). An example of such a system is a porous particle, with ions penetrating its bulk. The discussion below does not pertain to metallic (electronic) conductors. As stated earlier, most metallic conductors will behave as non-conducting particles in an electrolyte solution. Dukhin and Derjaguin (1974) give a full discussion on the effect of the particle conductivity. The expression for the case of a spherical particle with a finite conductivity is given by U=

2ǫζ E∞ F (κa, K ′ ) 3µ

(9.91)

σ′ σ∞

(9.92)

where K′ = and F (κa, K ′ ) = 1 + 2

(1 − K ′ ) [f (κa) − 1] (2 + K ′ )

(9.93)

Here σ ′ and σ ∞ are the electric conductivities of the particle and bulk electrolyte medium, respectively, and f (κa) is Henry’s function given by Eq. (9.85). For a non-conducting spherical particle σ ′ = 0 and K ′ = 0. This leads to 1 (1 − K ′ ) = (2 + K ′ ) 2 and F (κa, K ′ ) ≡ f (κa) Equation (9.91) then becomes U=

2 ǫζ E∞ f (κa) 3 µ

which is Eq. (9.84). For a perfectly conducting sphere, σ ′ → ∞ leads to K′ → ∞

and

(1 − K ′ ) → −1 (2 + K ′ )

Hence U=

2ǫζ E∞ [3 − 2f (κa)] 3µ

As f (κa) = 1 for κa → 0 (large Debye length) the above equation indicates that [3 − 2f (κa)] → 1. Consequently, for κa → 0, both the conducting and non-conducting spheres behave in a similar manner. This is quite reasonable as, for a large Debye length, κ −1 , the electric field is undisturbed by the presence of a particle. However, for

9.3 IMPROVED SOLUTIONS: ARBITRARY DEBYE LENGTH

321

κa → ∞ the situation is different for the conducting sphere. Here f (κa) → 3/2 as κa → ∞ and [3 − 2f (κa)] → 0. Hence U → 0 for κa → ∞ (small Debye length). For the case of an arbitrary particle shape having K ′ = 1 (i.e., σ ′ = σ ∞ ) (1 − K ′ ) =0 (2 + K ′ ) leading to F (κa, 1) = 1. This is Hückel’s model where the electric field is distorted by the double layer but not by the presence of the particle. For the case of non-conducting cylinders aligned parallel to the electric field, the electric field E∞ is always parallel to the particle surface for all κa values. Here Smoluchowski’s model is valid: i.e., F = 3/2 for all κa values. Gorin (1939) extended Henry’s solution for the case of non-conducting cylinders oriented normal to the electric field. The F values are given in Figure 9.8. The above analysis includes all Henry’s assumptions cited earlier. It should be remembered, however, that due to polarization, all particles act as non-conductors and hence having K ′ = 0 is an academic curiosity (Hunter, 1981). For randomly oriented cylinders, Keizer et al. (1975) showed that Ueffective =

 1 U + 2U⊥ 3

(9.94)

Figure 9.8. Values of Henry’s function for particles of various shapes and conductivities. (a) Non-conducting sphere (K ′ = 0); (b) conducting sphere (K ′ = ∞); (c) and (d) nonconducting cylinder (K ′ = 0) with axis parallel and perpendicular to the applied field, respectively; (e) particle of any shape with K ′ = 1. Here K ′ = σ ′ /σ ∞ , i.e., the ratio of the particle conductivity to the electrolyte solution conductivity.

322

ELECTROPHORESIS

where U and U⊥ are the electrophoretic mobilities of cylinders parallel and normal to the electric field, respectively. For more details on electrophoresis of cylinders and arbitrary shape particles, the reader can refer to Hunter (1981), Morrison (1970), and Van der Drift et al. (1979). 9.3.4

Alternate Forms of the Electrophoretic Velocities

The solutions given so far are expressed in terms of a constant potential at the particle surface. We shall now discuss the case of electrophoretic velocity of spherical particles expressed in terms of a constant surface charge. For a particle carrying a surface charge Qs in an electric field E∞ , the electrical force is given by F E = Qs E ∞

(9.95)

Equating this force with the drag force on the particle, Eq. (9.33) and using Eq. (9.29), we obtain Qs E∞ (1 − κa) for κa ≪ 1 (9.96) U= 6π aµ This is the electrophoretic velocity of a particle with constant surface charge in the Hückel limit (κa ≪ 1). For the case of a thin electric double layer, κa ≫ 1, the electrophoretic velocity of the particle is given by the Helmholtz–Smoluchowski equation, (9.41). Now, using Eq. (9.29), the surface charge on the particle can be related to its zeta potential at large κa as Qs = 4π a 2 ǫζ κ

for κa ≫ 1

(9.97)

Combining Eq. (9.41) with Eq. (9.97) to eliminate ζ , one obtains U=

Qs E∞ 4π a 2 µκ

for κa ≫ 1

(9.98)

Equations (9.96) and (9.98) are the Hückel and Helmholtz–Smoluchowski expressions for constant surface charge, respectively. Following the non-dimensionalization given by Russel et al. (1989), we can write β∗ =

qs∗ =

aeE∞ kB T

electric field strength

eQs 4π aǫkB T

surface charge density

(9.99)

(9.100)

and µa U = ǫ ∗



e kB T

2

U

electrophoretic velocity

(9.101)

9.3 IMPROVED SOLUTIONS: ARBITRARY DEBYE LENGTH

323

Making use of Eqs. (9.99) to (9.101), the dimensionless electrophoretic velocity U ∗ can be given for the constant surface charge boundary condition as Constant surface charge: U∗ =

2 ∗ ∗ β qs 3

for κa ≪ 1

(9.102)

and

qs∗ for κa ≫ 1 (9.103) κa Putting ζ ∗ = eζ /kB T , the electrophoretic velocities corresponding to Eqs. (9.34) and (9.41) are given for the constant surface potential condition in dimensionless form as U ∗ = β∗

Constant surface potential: U∗ =

2 ∗ ∗ β ζ 3

for κa ≪ 1

(9.104)

and U ∗ = β ∗ζ ∗

for κa ≫ 1

(9.105)

It is possible to generalize the limiting cases given above in terms of Henry’s solution for both constant surface potential and surface charge. The dimensionless form of Eq. (9.29) is qs∗ = ζ ∗ (1 + κa)

(9.106)

Making use of the above expression, Henry’s solution becomes (Russel et al., 1989), Constant surface charge: U∗ =

∗ ∗

2 β qs f (κa) 3 1 + κa

(9.107)

2 ∗ ∗ β ζ f (κa) 3

(9.108)

Constant surface potential: U∗ =

Figure 9.9 shows the scaled electrophoretic mobility (3U ∗ /2β ∗ qs∗ ) for the case of a constant surface charge and (3U ∗ /2β ∗ ζ ∗ ) for the case of a constant surface potential. It is evident from the figure that changes in κa affect the scaled electrophoretic mobility when either the charge or the potential is held constant. While the Hückel limit (κa ≪ 1) is unaffected by the electrical boundary condition on the particle surface, the scaled electrophoretic mobilities in the Helmholtz–Smoluchowski limit (κa ≫ 1) are vastly different for the two types of boundary conditions. In the case of constant surface potential, the scaled electrophoretic mobility stays non-zero throughout the

324

ELECTROPHORESIS

Figure 9.9. Variation of the scaled electrophoretic mobility with κa for constant potential and constant surface charge particles based on Henry’s solution.

range of κa because as the double-layer thickness diminishes, the particle charge increases. In case of constant charge particles, however, the scaled electrophoretic mobility becomes vanishingly small at large κa values. It should be remembered that although we treated the case of holding the surface charge or the potential constant, the particle charge or its potential and the diffuse-layer thickness are closely related to each other in an experimental situation. For a given particle, changes in ionic strength alter the particle charge as well as the diffuse-layer thickness (Russel et al., 1989).

9.3.5

Solutions Accounting for Relaxation Effects

Henry’s solution did not consider the effects arising from the distortion of the electrical double layer around a charged particle as it migrates along the imposed electrical field (cf., Figure 9.2). Strictly speaking, the convective relaxation effect was not accounted for in Henry’s solution. Since the charged ions are also dynamic in presence of the electrical field, the double layer becomes distorted. The ions move under the coupled influence of the external electrical field as well as the field due to the charged particle to restore the shape of the electric double layer. The ionic movement occurs due to convection, diffusion, and migration. Of these three modes, only the diffusionmigration based ion movements are considered in Henry’s result for low particle surface potentials. The asymmetry in the electrical double layer also gives rise to a convective migration of the ions in the vicinity of the particles. This process of relaxation becomes very important at higher particle surface potentials.

9.3 IMPROVED SOLUTIONS: ARBITRARY DEBYE LENGTH

325

The early works of Overbeek (1943) and Booth (1950) based on perturbation techniques provided some insight into the relaxation problem, although these solutions were limited to low surface potentials owing to the complexity of the problem. Wiersema et al. (1966) provided a numerical technique to solve the governing mobility equations for a binary electrolyte. Their study provided tabulated information on the mobility of a particle with different zeta potentials. However, their iterative numerical scheme failed to converge for zeta potentials greater than 150 mV for a 1 : 1 electrolyte. A more robust numerical scheme to solve the governing mobility equations was provided by O’Brien and White (1978). All of the above studies start from a set of perturbation equations pertaining to the mobility of the particle and the ions in an applied electrical field as outlined in Section 9.3.1. For the general case of arbitrary Debye length and surface potential, O’Brien and White (1978) developed a computational scheme to solve the governing perturbation equations (Section 9.3.1) for the electrophoretic problem corresponding to a constant surface potential on the particle. Figures 9.10 and 9.11 summarize the solutions given by O’Brien and White. Figure 9.10 shows the variation of dimensionless electrophoretic mobility with the dimensionless zeta potential for small and large κa. Employing Eqs. (9.99) and (9.101), the dimensionless electrophoretic mobility is written as 3µe 3U ∗ 3µe U η∗ = = η= 2ǫkB T E∞ 2ǫkB T 2β ∗

Figure 9.10. Dimensionless electrophoretic mobility of a spherical particle in KCl. Left: κa < 2; Right: κa > 3 (O’Brien and White, 1978).

326

ELECTROPHORESIS

Figure 9.11. Effect of counterion valence on dimensionless electrophoretic mobility of a spherical particle at κa = 5 (O’Brien and White, 1978).

It is observed from Figure 9.10 that for eζ /kB T = ζ ∗ ≤ 2, for κa → 0, the slope of the dimensionless electrophoretic mobility, 3U ∗ /2β ∗ , versus ζ ∗ is unity as would be expected for Eq. (9.104). For κa → ∞, the slope given by Figure 9.10 is 3/2 in accordance with Eq. (9.105). Deviation from Henry’s solution occurs for ζ ∗ ≥ 2 where the dimensionless electrophoretic mobility, 3U ∗ /2β ∗ , versus ζ ∗ curves are no longer linear. For 0.01 < κa < 2.75, the increase in 3U ∗ /2β ∗ with ζ ∗ becomes more gradual and for κa ≈ 2, 3U ∗ /2β ∗ becomes nearly independent of ζ ∗ . In other words, the particle mobility becomes unaffected by its surface potential. For κa > 3, the dimensionless electrophoretic mobility 3U ∗ /2β ∗ has a maximum value which becomes particularly pronounced at high κa values. This maximum value occurs at ζ ∗ ∼ = 5 − 7, which corresponds to ζ ∼ = 0.125 − 0.175 V for a (1 : 1) electrolyte solution. Dukhin and Semenikhin (1970), using a perturbation analysis, also showed the presence of maxima in the electrophoretic mobility arising from the convective relaxation effects. Figure 9.11 shows effects due to the valence of the electrolyte. The electrolyte solutions correspond to (1 : 1): KCl, (2 : 1): Ba(NO3 )2 , and (3 : 1): LaCl3 . The maximum in the dimensionless mobility moves from ζ ∗ ∼ = −5 for the (1 : 1) electrolyte to ζ ∗ ∼ = −1.7 for the (3 : 1) electrolyte. = −2.5 for the (2 : 1) electrolyte and to ζ ∗ ∼ These values are in the ratios of 1 : 1/2 : 1/3, which are inversely proportional to the valencies. From Figure 9.10, it is apparent that the maxima exhibited by the dimensionless electrophoretic mobility 3U ∗ /2β ∗ above (eζ /kB T ) ∼ 3 introduces difficulty in the estimation of U ∗ . This is owing to the fact that same value of the mobility is obtained for two different zeta potentials. This is shown schematically in Figure 9.12 where two possible values for ζ ∗ exist.

9.4 ELECTROPHORETIC MOBILITY IN CONCENTRATED SUSPENSIONS

327

Figure 9.12. Evaluation of dimensionless electrophoretic mobility at large values of ζ potential.

The solution given by O’Brien and White (1978) takes into consideration the relaxation, the electrophoretic retardation, and high surface potential effects. However, this solution still applies for a single colloidal particle in an infinitely large electrolyte reservoir, in other words, in an infinitely dilute colloidal suspension. The results of O’Brien and White cannot be applied for concentrated colloidal suspensions, since the solution does not account for the presence of the neighboring particles. In the next section, we will discuss approaches for modelling electrophoretic mobility of a swarm of particles representing a colloidal suspension with a finite particle concentration i.e., particle volume fraction.

9.4 ELECTROPHORETIC MOBILITY IN CONCENTRATED SUSPENSIONS Under most practical circumstances, measurements of electrophoretic mobility, or conversely, electroosmotic flow of an electrolyte through a bed of stationary colloidal particles must consider the effects of numerous charged colloidal particles. A colloidal suspension with a finite concentration of particles will clearly modify the electrophoretic mobility of a particle from the mobilities computed in the limit of infinite dilution, as discussed in the previous section. This difference in mobilities is affected by two factors. First, the presence of neighboring particles affects the fluid velocity field in the immediate vicinity of each particle. Secondly, the proximity of the charged particles will modify the ion distribution, and hence, the structure of the electrical double layer surrounding each colloidal particle. It is therefore important to know how the particle concentrations affect the electrophoretic mobilities of the particles. Incidentally, the same mathematical construct used to obtain the electrophoretic

328

ELECTROPHORESIS

mobility of a particle in a suspension can be employed to determine the electroosmotic flow of an electrolyte through the interstices of a porous medium consisting of stationary charged colloidal particles, since electrophoresis and electroosmosis are simply two different manifestations of the relative velocity between the particle and the electrolyte solution. 9.4.1

Cell Models for the Hydrodynamic Problem

It is clearly discernable that a colloidal suspension representing a swarm of particles is generally unstructured, implying that the distance between any two neighboring particles will be randomly distributed. Rigorous analysis of such unstructured suspensions are often only possible through statistical mechanics. However, in colloid science literature, a class of simplified models called the cell models has been frequently employed with considerable success. The cell model makes a simplified assumption regarding the suspension structure. Instead of considering the overall structure, or arrangement of the particles in the suspension, these models focus on a single particle and a representative volume of the fluid phase enclosing the particle. The representative fluid volume surrounding each particle is determined from the overall volume fraction (or conversely, the porosity) of the swarm of particles in the suspension. Assuming spherical colloidal particles of radius a, the volume fraction of particles in the suspension, φp , is related to the particle number concentration, N0 (m−3 ), as φp =

4 3 π a N0 3

The porosity of the suspension, i.e., the volume fraction of the electrolyte solution in the suspension is given by f = 1 − φp The cell model for a suspension of spherical colloidal particles of radius a assigns a spherical fluid shell surrounding the sphere, where the radius of the shell is related to the particle volume fraction (or porosity) as a = φp1/3 = (1 − f )1/3 b

(9.109)

Here, b is referred to as the radius of the cell. Equation (9.109) simply assigns a radius b to the spherical cell such that the particle to cell volume ratio equals the particle volume fraction throughout the suspension. A typical cell model representation of a suspension of spherical colloidal particles is depicted schematically in Figure 9.13. Once a colloidal suspension is modelled in terms of a cell, one needs to describe the governing equations for fluid flow, electric potential distribution, and ionic fluxes in the cell to solve an electrokinetic problem. While the basic governing equations remain mostly unchanged for a cell compared to their infinite dilution counterparts (see Section 9.3), the boundary conditions at the outer cell surface (at radius b) become

9.4 ELECTROPHORETIC MOBILITY IN CONCENTRATED SUSPENSIONS

329

Figure 9.13. Schematic representation of cell model of a colloidal suspension. The colloidal dispersion (a) is represented as a cluster of cells (b). Each cell contains one particle surrounded by a fluid envelope (c). The radius of the envelope is determined from the particle volume fraction in the suspension.

the most important factors determining the suitability and accuracy of the model. Specification of this boundary condition in different manners can lead to different results. For the following discussion, we will consider a spherical coordinate system (r, θ, φ). Let us first consider the fluid mechanics part of the problem. The Stokes equation, applicable for low Reynolds number flow, for the relative motion of the fluid with respect to the particle (assuming a particle fixed coordinate system) is µ∇ 2 u = ∇p + fb

(9.110)

where u is the fluid velocity vector, fb is the body force per unit volume, which may contain the gravitational force (ρg) and the electrical body force, ρf ∇ψ. The Stokes equation needs to be solved along with the continuity equation ∇ · u = 0 to obtain the velocity distribution in the cell. Employing the vector identity ∇ 2 u = ∇(∇ · u) − ∇ × ∇ × u

330

ELECTROPHORESIS

we can rewrite the Stokes equation as (9.111)

µ∇ × ∇ × u + ∇p = −fb

Use of different body forces in the above equation leads to different classes of problems. For instance, when only the gravitational force is considered in Eq. (9.111), the cell model pertains to the problem of hindered settling (sedimentation) of an uncharged colloidal suspension. When the body force is solely comprised of the electrical force, the cell model pertains to electrophoresis. Finally, when the body force contains both gravity and electrical forces, the problem is that of sedimentation of charged particles in a gravitational field. Of course, the problems of electrophoresis and sedimentation of charged particles will require solution of additional equations for the electric potential and the ionic fluxes. One now needs to provide the appropriate boundary conditions for the Stokes equation, (9.111), governing the fluid mechanical problem. Once again, we will assume that the particle is stationary and the fluid flow in the cell is U = −U ix . Figure 9.14 depicts the relevant geometry. The fluid boundary conditions at the particle surface will be that of zero slip, given by ur (r, θ) = uθ (r, θ) = 0

at r = a

(9.112)

At the cell boundary, r = b, one can define the radial velocity in terms of the mean relative velocity of the bulk fluid with respect to the particle, U , as ur (r, θ) = n · u = −U cos θ

at r = b

(9.113)

where n is the unit outward normal to the spherical cell surface (n = ir ). The final boundary condition at the outer boundary is stated differently in different cell models. Happel (Happel, 1958; Happel and Brenner, 1965, 1983) suggested that a plausible condition at the outer boundary is the absence of any tangential shear stress, which can be written as

∂(uθ /r) 1 ∂ur τrθ = µ r + =0 ∂r r ∂θ

at r = b

(9.114)

When the governing fluid dynamic problem is solved with this set of boundary conditions, we refer to it as the Happel cell model. An alternative approach of defining the boundary condition at r = b was provided by Kuwabara (1959), who stated that the azimuthal vorticity be zero at the outer cell envelope: ωθ =

1 ∂(ruθ ) 1 ∂ur − =0 r ∂r r ∂θ

at r = b

(9.115)

9.4 ELECTROPHORETIC MOBILITY IN CONCENTRATED SUSPENSIONS

331

Figure 9.14. Geometrical considerations used for assigning boundary conditions for the cell model. In a particle fixed reference frame, the fluid velocity is U along the negative x direction at the outer cell boundary r = b. Happel cell model imposes an outer cell surface boundary condition employing the tangential shear stress, while Kuwabara cell model imposes a boundary condition based on the vorticity.

A general vectorial notation of Kuwabara’s boundary condition is ω =∇ ×u=0

at r = b

(9.116)

When the fluid mechanics problem is solved with the Kuwabara boundary condition, the model is referred to as the Kuwabara cell model. Using the solution of a given cell model, the drag on a particle within the cell is calculated, and its hindered settling velocity is then compared with Stokes free settling velocity. The term hindered settling velocity is used for the settling velocity of a suspension. The word “hindered” refers to the lowering of the settling velocity of a particle in a suspension due to the presence of neighboring particles. When one solves the fluid mechanics problem for the hindered settling of a monodisperse uncharged colloidal suspension (that is, in absence of any electrical body force), Happel and Kuwabara cell models yield slightly different expressions for the velocity field and the hindered settling velocities. For instance, the ratio of the hindered settling velocity to the free settling velocity (Stokes settling velocity) obtained by solving the above cell model using the Happel boundary condition (stress

332

ELECTROPHORESIS

free cell surface) yields 1/3

Happel

UH S USt

=

5/3

1 − 23 φp + 23 φp − φp2 5/3

1 + 23 φp

where UHS and USt refer to the hindered and Stokes free settling velocities, respectively. The corresponding expression for the settling velocity ratio obtained using the Kuwabara boundary condition (zero vorticity at the cell boundary) is Kuwabara UHS 9 1 = 1 − φp1/3 + φp − φp2 USt 5 5

The results from these two cell models are depicted in Figure 9.15. One can note from this figure that the two results are usually within a few percent of each other over a wide range of volume fractions (0 < φp < 0.74), and are in excellent agreement at very low (φp → 0) as well as very high (φp > 0.5) volume fractions. This comparison provides a benchmark regarding the extent by which the predictions of the Kuwabara and Happel cell models can differ in terms of the fluid mechanical problem in absence of any electrical effects. In the following, we discuss cell model based theories of electrophoretic mobility in concentrated particle suspensions.

Figure 9.15. Comparison of the hindered settling velocity of uncharged monodisperse spherical particles at different volume fractions obtained employing the Happel and Kuwabara cell models.

9.4 ELECTROPHORETIC MOBILITY IN CONCENTRATED SUSPENSIONS

333

9.4.2 The Levine–Neale Cell Model It is reasonable to state that in a regime where the double-layer thickness is very small relative to the particle size, i.e., κa ≫ 1, one would expect that the electrophoretic velocity of a given particle is not affected by the presence of other particles, especially at low particle concentrations. Here, the particle electrophoretic mobility U/E∞ would not be concentration dependent. For the case of κa ≫ 1, one would expect Smoluchowski’s equation to hold (Reed and Morrison 1976; Hunter 1981). For the case of either very high particle concentrations or small κa, the double layers of adjacent particles overlap and one would not expect Henry’s equation for a single particle to hold. Moreover, for κa ≪ 1, one would also expect Hückel’s equation to be inappropriate even for fairly dilute suspensions. In order to study the effect of particle concentration on the electrophoretic mobility of a particle, it is necessary to model the particle suspension accounting for the presence of neighboring particles. Among the early models is that of Möller et al. (1961). They developed a model which led to an expression for the variation of electrophoretic mobility of monodisperse spheres of uniform and constant surface charge as the double layers overlap. In this model, each particle is enclosed by a spherical shell of the electrolyte medium which contains just the right amount of excess ions to neutralize the total charge on the surface of the particle. Möller et al. (1961) gave an expression for the electrophoretic mobility of a particle in a suspension of the form η=

U ǫζ = · g1 (κa, φp ) E∞ µ

where g1 (κa, φp ) is a function of κa and φp , µ is the fluid viscosity, and η is the electrophoretic mobility of a particle. The limits of the applicability of the above expression is given by Levine and Neale (1974b), who provide the maximum values of κa corresponding to a given particle volume fraction for which the mobility expression of Möller et al. (1961) is valid. This is shown in Table 9.2 Levine and Neale (1974a) extended Henry’s solution and the model given by Möller et al. (1961) for the case of suspension of spherical particles employing both the Happel and Kuwabara cell models. Henry’s theory (1931) for the electrophoresis of a single isolated sphere was used. The sole modification in this regard was that the hydrodynamic and electrical boundary conditions at r → ∞ in Henry’s approach was replaced by appropriate conditions at the outer shell boundary r = b in the Levine and Neale approach. A brief outline of their cell model is given below.

TABLE 9.2. Range of Validity of the Electrophoretic Mobility Expression of Möller et al. (1961) According to Levine and Neale (1974b). φp κa

0.0001 0.5

0.001 0.7

0.01 1.2

0.05 2.3

0.2 5

0.4 8

0.6 15

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ELECTROPHORESIS

As in Henry’s approach, the total potential is assumed to be (φ + ψ), where ψ arises from the charged particle3 and φ from the applied field. The following equations are solved: ǫ∇ 2 ψ = −ρf (9.117) where ψ = ψ(r), i.e., ψ depends on radial distance only, and ρf is the free charge density. The boundary conditions are dψ qs =− dr ǫ

at r = a

(9.118)

and dψ = 0 at r = b (9.119) dr where qs is the surface charge density. Boundary condition (9.118) relates the ψ gradient to the surface charge density qs . As an equivalent to this boundary condition one can use ψ = ζ at r = a, see Eqs. (9.28) and (9.73) relating the zeta potential to the surface charge. The boundary condition (9.119) indicates that there is no current entering the particle; i.e., the surface is non-conducting or that the cell is isolated from other cells. The potential due to the external field φ(r, θ) is governed by4 ∇ 2φ = 0

(9.120)

with the boundary conditions ∂φ =0 ∂r

at r = a

(9.121)

and ∂φ = −E∞ cos θ ∂r

at r = b

(9.122)

Making use of the Debye–Hückel approximation, the solutions of the potentials are given by  q a  a   sinh(κb − κr) − κb cosh(κb − κr) s ψ(r) = ǫ r (1 − κ 2 ab) sinh(κb − κa) − (κb − κa) cosh(κb − κa)

(9.123)

3

Once again, in this section, we denote the stationary electric double layer potential as ψ. Recall the conceptual inconsistency with this approach pointed out while presenting Henry’s calculations. Once again, we simply reproduce the approach originally presented by Levine and Neale (1974a), noting that the application of these equations do not lead to a correct description of the electrical potential distribution around the particle. However, the final results for the electrophoretic mobility turn out to be accurate. 4

335

9.4 ELECTROPHORETIC MOBILITY IN CONCENTRATED SUSPENSIONS

and φ(r, θ) = −

  E∞ a3 r + 2 cos θ (1 − αp ) 2r

(9.124)

As stated earlier, the fluid flow around the spherical particle is governed by the Stokes equation for creeping flows. In the present electrophoresis problem, the body force in Eq. (9.110) is given by ρf ∇(ψ + φ), which yields µ∇ 2 u = ∇p + ρf ∇(φ + ψ)

(9.125)

with the continuity equation ∇ ·u=0

(9.126)

The boundary conditions are given by Eqs. (9.112), (9.113), and, depending on whether Happel or Kuwabara cell models are employed, either Eq. (9.114) or (9.115), respectively. The solution of Eqs. (9.125) and (9.126) subject to the boundary conditions Eqs. (9.112), (9.113), and (9.115) was given by Levine and Neale (1974a). The electrophoretic mobility, η, was defined as U η= = E∞



ǫζ µ



g2 (κa, φp )

Making use of a force balance similar to Henry’s approach (Section 9.3.2), the function g2 (κa, φp ), which accounts for the presence of neighboring particles, was evaluated employing the Kuwabara cell model (Levine and Neale, 1974a). The function g2 was derived assuming zeζ /kB T < 1 (Debye–Hückel limit). It was of the form g2 (κa, φp ) =

2 [−I1 + I2 ] 3(1 − y 3 )

(9.127)

where

5Q Ry Ry 3 (κa)4 (κa)2 − − + (κa)3 − I1 = − 1 + Rκa + 16 48 4 12 96

 Ry (κa)6 Q (κa)4 − − − (κa)5 + I 96 48 8 96 

and

R 3 3Q R Q (κa)2 + + + − (κa) − (κa)2 (κa) y3 10 10 40 



3 4 Q Ry Ry Q (κa) (κa)6 + − + − I (κa)3 − (κa)5 − 120 30 240 240 120 240

 I2 = − 1 +

336

ELECTROPHORESIS 1/3

In the above expressions, y = φp = a/b, 1 = sinh(κb − κa) − κb cosh(κb − κa) R

Q=

1 − κb tanh(κb − κa) tanh(κb − κa) − κb

and I=



κb κa

A cosh t − B sinh t dt t

The parameters A and B are expressed as A = R[sinh(κb) − κb cosh(κb)] B = R[cosh(κb) − κb sinh(κb)] One should note that the evaluation of the integral I is non-trivial for large values of κa > 10. In that case, one should express the integral I in terms of the exponential integrals. It is sometimes more convenient to use an alternative form of the Levine– Neale expression for the electrophoretic mobility, given by (Ohshima, 1997a) ǫζ 2 3 µ(1 − φp )  b  a 3 3  a 5 φ   r 3  a 5  1 dψ (0) p 1− × + dr 1 − 10 + −6 r 2 r 10 ζ dr a r a (9.128)

η=−

where the function ψ (0) is given by ψ

(0)

a =ζ r



sinh[κ(b − r)] − κb cosh[κ(b − r)] sinh[κ(b − a)] − κb cosh[κ(b − a)]



(9.129)

The function g2 (κa, φp ) obtained using the Kuwabara cell model is plotted in Figure 9.16. For φp = 0, g2 becomes (2/3)f (κa), where f (κa) is Henry’s function. The plot for g2 (κa, φp ) against κa for various particle volume fractions φp clearly shows that at κa > 102 , the volume fraction has little influence on the electrophoretic mobility, i.e., g2 (κa, φp ) ∼ = 1.0, and the Smoluchowski equation can be used. For κa ≪ 1, g2 (κa, φp ) becomes 2/3 only when φp = 0. In this limit of small κa, the particle concentration has a marked effect on g2 (κa, φp ). The Kuwabara cell model has been preferred over the Happel cell model following the work of Levine and Neale (1974a), who noted that the Happel cell model predicts that the electrophoretic velocity U slightly increases as the volume fraction

9.4 ELECTROPHORETIC MOBILITY IN CONCENTRATED SUSPENSIONS

337

Figure 9.16. Variation of the function g2 (κa, φp ) with κa for different particle volume fractions.

of the particles increases. This behavior is considered counter intuitive and it is not observed experimentally. On the other hand, as noted from Figure 9.16, Kuwabara cell model predicts that the electrophoretic velocity corresponding to large κa values is independent of volume fraction. Consequently, the Kuwabara cell model is generally employed in the analysis of electrophoresis in concentrated suspensions. For the case of high surface potentials, Kozak and Davis (1989a) used Kuwabara’s model to include the relaxation effect as well. They used the perturbation approach of O’Brien and White (1978) to obtain the potential distribution around the spherical particle, albeit with the far field (r → ∞) condition replaced by the appropriate boundary conditions at the cell surface r = b. Some of their results are shown in Figures 9.17 to 9.19 for large values of κa. Deviation from Smoluchowski’s equation is shown on these plots. Their dimensionless mobility is defined such that the dimensionless mobility for the Smoluchowski equation is equal to 2(zeζ /kB T ) where (zeζ /kB T ) is the dimensionless surface potential. In other words, the dimensionless mobility is defined by Kozak and Davis (1989a) as M∗ =

2µU ze 2U ∗ = ∗ ǫE∞ kB T β

The Smoluchowski equation in dimensionless form is then given by M ∗ = 2ζ ∗ Figure 9.17 shows Kozak and Davis work for φp = 0 (or porosity f = 1). For a single sphere, it compares well with O’Brien and White (1978) where high potential

338

ELECTROPHORESIS

Figure 9.17. A comparison among the analyses of O’Brien and White (1978), Smoluchowski (1918), and Kozak and Davis (1989a,b) for single spheres in a (z : z) electrolyte.

and relaxation effects are taken into account in the analysis. Figures 9.18 and 9.19 are given for f = (1 − φp ) in the range of 0.5 to 1.0. The deviation from the Smoluchowski analysis becomes even greater at a higher surface potential. The results of Kozak and Davis (1989a,b) when adjusted by a factor of 3/4, reduce to the results provided by Carrique et al. (2001) for the electrophoretic mobility. Various asymptotic expansion expressions are given by Kozak and Davis (1989b). It should be noted, however, that although the analysis was made using a higher potential, polarization usually occurs at a high potential leading to a variable dielectric constant ǫ. Moreover, the above models neglect the effect of a finite ion size. The above simplifications would lead to a greater deviation form Henry’s curve. For models dealing with a variable dielectric constant and finite size of ions, the reader is referred to Gur et al. (1978a,b), Pintauro and Verbrugge (1989), and Datta and Kotamarthi (1990). An experimental study on electrophoresis of porous aggregates was conducted by Miller and Berg (1993). They obtained reasonable agreement between the experimentally measured electrophoretic mobility of porous aggregates and the aforementioned theoretical analysis of Levine and Neale (1974a,b) and with the theoretical study of Miller et al. (1992). The theoretical analysis tends to underestimate the effect of the porosity of the aggregates. Although the Levine and Neale cell model provides accurate estimates of the electrophoretic mobility of a concentrated suspension for low surface potentials, there seems to be an apparent controversy regarding one of the boundary conditions used in their approach. Referring back to the boundary condition Eq. (9.122), we note that the

9.4 ELECTROPHORETIC MOBILITY IN CONCENTRATED SUSPENSIONS

339

Figure 9.18. The effects of porosity, f = 1 − φp , and zeta potential on the dimensionless mobility for κa = 50 (Kozak and Davis, 1989b).

Figure 9.19. The effects of porosity and zeta potential on the dimensionless mobility for κa = 150 (Kozak and Davis, 1989b).

340

ELECTROPHORESIS

electric field at the outer cell boundary (r = b) is given in terms of E∞ , which is the undisturbed external electric field. One may ask the question as to whether the external electric field exists at the cell boundary. In other words, does the electric field applied between two external electrodes bounding a colloidal dispersion remain unchanged at the boundary of a cell, which exists inside the swarm of particles representing the dispersion? We will revisit this problem regarding the cell surface boundary condition later in this chapter. 9.4.3 The Ohshima Cell Model Significant contribution has been made in the field of electrophoresis by Ohshima (Ohshima, 1997a,b; Ohshima, 1999; Ohshima, 2000). He employed the Kuwabara cell model to develop the theory of electrophoresis in concentrated suspensions based on a perturbation approach. Here, we first summarize the general construct of Ohshima’s model. Considering a spherical coordinate system (r, θ, φ) with one of the particle centers as the origin, and the imposed external electric field aligned parallel to the polar axis (θ = 0), the governing electrokinetic equations are written for the fluid flow, ionic fluxes, and the electrochemical potential. The system comprises of N mobile ion species with valence zi and drag coefficient λi = kB T /Di . The bulk concentration of the ions in the electroneutral solution is denoted by ni∞ . The governing electrokinetic equations underlying Ohshima’s model are: µ∇ × ∇ × u + ∇p = −ρf ∇ψ ∇ ·u=0 vi = u −

fluid continuity

1 ∇µi λi

∇ · (ni vi ) = 0 ρf =

N


Stokes equation

ionic velocity

(9.130) (9.131) (9.132)

conservation of ions

(9.133)

charge density

(9.134)

zi eni

i=1

µi = µ0i + zi eψ + kB T ln ni

electrochemical potential of ion

(9.135)

and ∇ 2ψ = −

ρf ǫ

Poisson equation

(9.136)

In the above equations, ni represents the local number concentration of the i th ionic species, µi is the chemical potential of the i th ionic species, and µ0i is the constant term in µi (the so-called reference chemical potential).

9.4 ELECTROPHORETIC MOBILITY IN CONCENTRATED SUSPENSIONS

341

Assuming that the electrical double layer around the reference colloidal particle in the spherical cell is only slightly distorted due to the applied electric field, the ionic concentration, ni , the electric potential, ψ, and the electrochemical potential, µi , can be represented as small perturbations from their equilibrium values: ni = n(0) i (r) + δni (r, θ)

(9.137)

ψ = ψ (0) (r) + δψ(r, θ)

(9.138)

µi =

µ(0) i (r)

+ δµi (r, θ)

(9.139)

where the superscript (0) refers to the equilibrium state and δ denotes a small perturbation from equilibrium. Employing Eq. (9.135), the perturbation variables δni , δψ, and δµi can be related as   δni δni δµi = zi eδψ + kB T ln 1 + (0) ≃ zi eδψ + kB T (0) (9.140) ni ni where the final expression in Eq. (9.140) is obtained using a Taylor expansion of the logarithmic term, ln[1 + δni /n(0) i ]. Note that the perturbation approach is similar to the one described in Section 9.3.1. In this case, however, the total electrochemical potential is employed as an additional perturbation parameter. The substitution of the perturbation expressions, Eqs. (9.137)–(9.139) in the governing equations leads to the following equations: µ∇ × ∇ × ∇ × u =

N


∇δµi × ∇n(0) i

(9.141)

i=1

and ∇·



n(0) i u

−

∇δµi n(0) i λi



=0

(9.142)

Furthermore, assuming that the concentration of the ions at equilibrium obeys Boltzmann distribution, the distribution of ψ (0) is obtained from the Poisson– Boltzmann equation 1 d r 2 dr



r

2 dψ

(0)

dr

N zi eψ (0) 1
=− zi eni∞ exp − ǫ i=1 kB T

(9.143)

Let us now define the boundary conditions for Eqs. (9.141) to (9.143). The boundary conditions at the particle surface are same as described in Section 9.3.1. These are zero slip, no ion penetration into the particle, and a known electric potential at the shear plane (or particle surface). The no-slip and no ion penetration boundary conditions are given by Eqs. (9.12) and (9.13) respectively. At the outer cell boundary, r = b, the boundary conditions for the Stokes equation are given by Eqs. (9.113)

342

ELECTROPHORESIS

and (9.116) following the Kuwabara cell model. The boundary condition for the electrostatic problem, Eq. (9.143) at the outer cell surface r = b was given by Ohshima (1997a) as dψ (0) =0 (9.144) dr

This condition is identical to the cell surface boundary condition employed by Levine and Neale (1974a), and simply states that the cell is electroneutral for the unperturbed (no external field) case. The stationary problem, Eq. (9.143), can be solved subject to the boundary conditions of a constant surface potential at r = a and Eq. (9.144) at r = b. It should be noted that although the electrophoretic velocity U appears in the boundary conditions, it is actually unknown. Hence, one needs to specify another boundary condition for the perturbation problem. This equation follows from the assumption that the net force on a cell must be zero. In other words, for the electrophoresis problem, the total electrical force applied to the dispersed system is zero, and accordingly, the hydrodynamic force acting on the cell must be zero. This condition is given by  = FH = τ · n dS = 0 (9.145) S

where the integral is evaluated over the surface S representing the outer cell boundary = r = b, and τ is the hydrodynamic stress tensor with the following components: ∂ur τrr = −p + 2µ ∂r   ∂uθ uθ 1 ∂ur + − τrθ = µ r ∂θ ∂r r

Note that pressure is specified in this formulation through the boundary condition, Eq. (9.145). The electrophoretic velocity U is implicitly embedded in the hydrodynamic stress. The final boundary condition at r = b involves specifying the electrochemical potential perturbation. Using the specification of the boundary condition originally given by Levine and Neale (1974a) for the potential gradient, ∇δψ · n = −E∞ · n = −E∞ cos θ

at r = b

(9.146)

as well as, using ∇δni · n = 0

at r = b

the electrochemical potential perturbation at the cell boundary can be written as ∇δµi · n = −zi eE∞ · n = −zi eE∞ cos θ

at r = b

(9.147)

343

9.4 ELECTROPHORETIC MOBILITY IN CONCENTRATED SUSPENSIONS

Note that Eqs. (9.146) and (9.147) are alternate but equivalent statements of the outer cell surface electrochemical boundary condition, with the only difference that the latter equation uses the electrochemical potential instead of the electric potential. In subsequent discussions, we will refer to either of these boundary conditions as the Levine–Neale boundary condition. The perturbation analysis of Ohshima (1997a) involves solving the governing perturbation equations, Eqs. (9.141) and (9.142), using the above boundary conditions. The solution of the perturbation problem yields the electrophoretic mobility (Ohshima, 1997a) 2ǫζ U = η(κa, φp ) = η = E∞ 3µ



a

b



 1 a3 P1 (r) 1 + dr + Q1 (b) 2 r3

(9.148)

where

  2r 3 a3 2 r3 3r 5 1 dψ (0) 3r 2 (κa)2 P1 (r) = − − 3+ 5 1− 2 + 3 − 3 6(1 − φp ) a a b 5 a 5a ζ dr and Q1 (b) =





(κa)2 a3 9b2 a3 b3 ψ (0) (b) 1 + 3 1+ 3 − 2 − 3 3ζ (1 − φp ) 2b a 5a 5b

Here the unperturbed electrical potential based on the solution of the Poisson– Boltzmann equation (9.143) is ψ (0) (r) = ζ

a κb cosh[κ(b − r)] − sinh[κ(b − r)] r κb cosh[κ(b − a)] − sinh[κ(b − a)]

Equation (9.148) requires numerical evaluation of the mobility. It is similar to the Levine and Neale (1974a) electrophoretic mobility expression. Equation (9.148) is valid for low ζ -potentials, which allows the linearization of the Poisson–Boltzmann equation required to arrive at the given expression. Although both solutions are for constant surface potential on the particles, the Levine–Neale solution uses the surface charge density boundary condition on the sphere, while the Ohshima approach directly uses the surface potential condition on the particle. This leads to slightly different forms for the electrical potential expression in the two solutions. For low surface potentials and weak overlap of the electrical double layers, Ohshima (1997a) derived an approximate analytical expression for the electrophoretic mobility from Eq. (9.148), which is in excellent agreement with the Levine and Neale (1974a) result, (Ohshima, 1997a). The expression for the mobility is (Ohshima, 1997a) ǫζ η= G(κa, φp ) (9.149) µ

344

ELECTROPHORESIS

where G(κa, φp ) =

    2 1 1+ M (κa, φ ) + M (κa, φ ) 1 p 2 p 3 2[1 + δ/(κa)]3

(9.150)

In the above equation, 3φp (1 + κaQ) (κa)2 + M1 = 1 − 2/3 (κa)2 (1 − φp ) 5P φp



1/3

1 − φp 3 3φp − + 1+ 3 1 − φp 1 − φp



  4/3 2(κa)2 (1 + φp /2) 1 9 φp 1/3 M2 = φp + 2/3 − 1/3 − 9P (1 − φp ) 5 φp 5φp P = cosh[κ(b − a)] − (κb)−1 sinh[κ(b − a)]   φp1/3   = cosh κa(φp−1/3 − 1) − sinh κa(φp−1/3 − 1) (κa)

and

  −1/3 −1/3 1 − κaφp tanh κa(φp − 1) 1 − κb tanh[κ(b − a)] Q= =   −1/3 −1/3 tanh[κ(b − a)] − κb tanh κa(φp − 1) − κaφp δ=

2.5 1 + 2 exp(−κa)

The interesting feature of this analytical expression is that although it was derived assuming weak overlap of double layers, it is in excellent agreement with the Levine and Neale (1974a) solution even for κa < 1, or substantial double layer overlap. Figure 9.20 shows a comparison of the dimensionless electrophoretic mobility, ηµ/ǫζ , where ǫζ /µ is the Helmholtz–Smoluchowski electrophoretic mobility, obtained using the expression of Levine and Neale (1974a) and Eq. (9.150) for selected values of the particle volume fraction, φp . 9.4.4

Suspension Electric Conductivity

Ohshima also provided an expression for the suspension electrical conductivity in terms of the particle volume fraction (Ohshima, 1999). The suspension conductivity, σ susp , is a function of the particle volume fraction, and differs from the bulk electrolyte conductivity. The bulk electrolyte conductivity is given by σ∞ =

N
i=1

zi2 e2 ni∞ /λi

9.4 ELECTROPHORETIC MOBILITY IN CONCENTRATED SUSPENSIONS

345

Figure 9.20. Variation of the dimensionless electrophoretic mobility with κa obtained using Ohshima’s approximate expression, Eq. (9.150) compared with the results of Levine and Neale (1974a).

where the Stokes–Einstein friction factor, λi , is defined as λi =

kB T Di

Note that we denoted the Stokes–Einstein friction factor as fi in Chapter 6, cf., Eq. (6.39). Ohshima’s expression for the suspension conductivity applies for low surface potentials and non-overlapping double layers. It is given by σ susp where

  N 3

z n /λ 1 − φ eζ i∞ i p i=1 i = σ∞ 1 − 3φp L(κa, φp ) N 2 1 + φp /2 kB T i=1 zi ni∞ /λi

1 L(κa, φp ) = − 3 3a ζ (1 − φp )(1 + φp /2)



a

b



a3 + r3 2

(9.151)



a 3 dψ (0) dr 1− 3 r dr

An approximate expression for L(κa, φp ) was also provided by Ohshima (1999), and it is given as   1 + κaQ 1 L(κa, φp ) = 1+ (κa)2 (1 − φp )(1 + φp /2) 2[1 + δ/(κa)]3

346

ELECTROPHORESIS

Here Q and δ are the same functions used in Eq. (9.150). The limiting form of Eq. (9.151) for uncharged particles (ζ = 0), as well as, for infinitesimally thin double layers, κa → ∞, is 1 − φp σ susp = ∞ σ 1 + φp /2

(9.152)

Further simplification of Eq. (9.152) yields for very dilute suspensions, φp → 0 (Russel et al., 1989),   σ susp 3 φ = 1 − (9.153) p σ∞ 2 9.4.5 The Shilov–Zharkikh Cell Model A large body of studies on electrophoresis is based on the Shilov–Zharkikh cell model (Shilov et al., 1981; Carrique et al., 2001). The basic construct of the model is identical to the perturbation approach delineated so far in this section. A major point of departure in the Shilov–Zharkikh model is the application of the boundary condition at the outer cell surface (r = b), which is given by: δψ = −bEsusp · n = −bEsusp cos θ

at r = b

(9.154)

The boundary condition of Eq. (9.154) is different from the outer cell surface boundary conditions used in the Levine–Neale cell model and the Ohshima cell model [given by Eqs. (9.122) and (9.146), respectively]. The Levine–Neale and Ohshima boundary conditions are identical, and in the following discussion, we will refer to both these conditions as the Levine–Neale boundary condition. There are two differences between the Shilov–Zharkikh boundary condition, Eq. (9.154), and the Levine–Neale boundary condition. First, the notation Esusp in Eq. (9.154) refers to a different electric field compared to the Levine–Neale boundary condition, which uses the external field, E∞ as in Eq. (9.146). Here, we will refer to Esusp as the internal electric field or the electric field within the suspension. Secondly, Eq. (9.154) specifies the perturbed potential, δψ at the outer cell boundary, whereas the Levine–Neale boundary condition specifies the gradient of the perturbed potential, ∇δψ, as in Eq. (9.146). Let us first address the difference between the internal and external electric fields. Figure 9.21 depicts a suspension between two external reservoirs of electrolyte solution. When a potential difference between the two external electrodes is set up, a current will flow between these electrodes. For a steady state process, the total current at every cross section in the volume between the two electrodes will be the same. The current densities (A/m2 ) at every cross section will also be the same, since the cross sectional area is the same everywhere. Denoting the spatial average current density vector inside the suspension as i and the current density in the external electrolyte solution as i∞ , we can therefore write i = i∞

(9.155)

9.4 ELECTROPHORETIC MOBILITY IN CONCENTRATED SUSPENSIONS

347

Figure 9.21. Schematic representation of electrophoresis in a concentrated suspension showing the internal and external electric field and conductivities. The total current I across any cross section, A, is constant, which implies that the current densities in the external electrolyte solutions and the suspension are the same.

Considering Figure 9.21, there is a slight difference in how the current flows through the suspension and through the external electrolyte. The current in the suspension is only due to the flowing electrolyte (the particles are non-conducting) and hence, in the suspension, the current passes only through the interstitial volume between the solid particles. This implies that the specific conductivity in the suspension, σ susp , is different from the specific conductivity of the external electrolyte solution, σ ∞ . One can apply Ohm’s law [see Eq. (6.72)] locally in the suspension and the external electrolyte reservoirs to relate the current density and the electric field. For the suspension, the relationship is i = σ susp Esusp

(9.156)

and for the external electrolyte, the corresponding relationship is i∞ = σ ∞ E∞

(9.157)

In Eqs. (9.156) and (9.157), Esusp and E∞ refer to the local electric field in the suspension and the external electric field, respectively. The local electric field in the suspension can be formally defined as the volume average of the potential gradient

348

ELECTROPHORESIS

within the suspension Esusp = −

1 Vsusp



∇ψ dVsusp = −∇ψ

(9.158)

Vsusp

where Vsusp is the volume of the suspension (which takes into consideration both the volume occupied by the particles as well as the volume of the electrolyte solution in the interstices). It is of interest to note that Eq. (9.158) is also used to define the sedimentation potential. The relationships between the current and the field given above are conceptually identical to what would be expected in an electrical circuit with resistances in series: The total current through each resistor is the same, although the potential difference across each resistor is different. Combining Eqs. (9.155), (9.156), and (9.157), one obtains σ susp Esusp = σ ∞ E∞

(9.159)

Denoting the unit vector along the direction normal to the electrode planes in Figure 9.21 as ix , one can obtain the magnitudes of the electric fields acting along the direction of ix as Esusp = Esusp · ix

and

E∞ = E∞ · ix

Using the above components of the field acting along the direction ix in Eq. (9.159), one can write σ susp Esusp = σ ∞ E∞ or Esusp σ∞ = susp E∞ σ

(9.160)

Referring to Eq. (9.151), which provides the relationship between the suspension conductivity and the external conductivity, one immediately observes that since the two conductivities are different, Eq. (9.160) would lead to a difference between the suspension and the external electric fields. Consequently, it is more appropriate to use the internal or suspension electric field, Esusp , to define the cell surface boundary condition. One should note, however, that as the suspension becomes dilute, φp → 0, the internal and external electric fields tend to become identical, Esusp → E∞ . We now turn our attention to the second difference between the Shilov–Zharkikh and Levine–Neale outer cell boundary conditions, namely, the use of the perturbed potential instead of the gradient of the perturbed potential at the outer cell boundary. Equation (9.154) was originally proposed by Shilov et al. (1981) based on an extensive analysis of the electrical properties of the cell. A simpler heuristic approach for rationalizing the use of the potential instead of the potential gradient in Eq. (9.154) is suggested here based on the approach of Kozak and Davis (1986). Consider the

9.4 ELECTROPHORETIC MOBILITY IN CONCENTRATED SUSPENSIONS

349

potential distribution obtained by Henry (1931), Eq. (9.63),   a3 φ = −E∞ r + 2 cos θ 2r Henry’s solution was obtained in the infinite dilution limit, φp → 0, where Esusp = E∞ . Using the above equation to determine the potential at the outer cell boundary, r = b, we obtain,   a3 φ = −Esusp b + 2 cos θ 2b

or

  1  a 3 φ = −b Esusp 1 + cos θ 2 b

(9.161)

φ = −bEsusp cos θ

(9.162)

Note that in applying Eq. (9.63) at the outer cell boundary, we have used the internal electric field in the suspension, Esusp . Equation (9.161), in the limiting case of a/b ≪ 1, reduces to The similarity between Eqs. (9.154) and (9.162) suggests that the use of the perturbed potential instead of the gradient of the perturbed potential leads to an appropriate representation of the boundary condition at r = b. Use of the Shilov–Zharkikh boundary condition in the cell model instead of the Levine–Neale boundary condition leads to an apparently different result for the electrophoretic mobility of a particle in a concentrated dispersion. Carrique et al. (2001) solved the cell model for electrophoresis employing both Levine–Neale and Shilov–Zharkikh boundary conditions. The expression for the electrophoretic mobility obtained by Carrique et al. (2001) employing the Shilov–Zharkikh boundary condition was  b

 U 1 a3 2ǫζ ηsusp (κa, φp ) ≡ ηsusp = H (r) 1 + = dr + F (r) Esusp 3µ 2 r3 a (9.163) where

  (κa)2 r3 2r 3 a3 2 3r 5 1 dψ (0) 3r 2 H (r) = − − 3+ 5 1− 2 + 3 − 3 6(1 + φp /2) a a b 5 a 5a ζ dr and

(κa)2 (0) 9b2 a3 b3 F (r) = ψ (b) 1 + 3 − 2 − 3 a 5a 5b 3ζ Note that in Eq. (9.163), the electrophoretic mobility of a particle in the suspension, ηsusp , is defined as U/Esusp based on the internal electric field, Esusp . The definition used in Eq. (9.163) is different from the definition of the mobility employed earlier in

350

ELECTROPHORESIS

this chapter, for instance in Eq. (9.148), which is based on the external electric field, η = U/E∞ . The expression for the mobility, ηsusp , obtained by Carrique et al. (2001), Eq. (9.163), differs from the mobility, η, obtained by Ohshima (1997a), Eq. (9.148), owing to a different dependence on the solute volume fraction. To emphasize this difference, we denote the mobility in Eq. (9.163) as ηsusp . There is considerable confusion in the electrophoresis literature, particularly among the studies pertaining to the calculation of electrophoretic mobility of concentrated suspensions, owing to somewhat misleading use of the two alternate definitions of the electric field, and hence, due to the use of the Shilov–Zharkikh and Levine– Neale boundary conditions. We note that the electric field used in the Levine–Neale boundary condition (also used in Ohshima’s model) was the external field, E∞ . However, it should have been the internal electric field Esusp . The implication of these two electric fields on the evaluated electrophoretic mobility will become clear once we consider the two commonly adopted definitions for the mobility. Smoluchowski’s original definition of electrophoretic mobility is given by η=

U · ix U = E∞ E∞ · ix

(9.164)

which is based on the external electric field, E∞ . Another definition of the electrophoretic mobility is possible, on the basis of the internal electric field, Esusp (Dukhin et al., 1999) U · ix U ηsusp = = (9.165) Esusp Esusp · ix This mobility can be termed as the internal or suspension mobility. The two definitions of the electrophoretic mobility, given by Eqs. (9.164) and (9.165) are perfectly legitimate and are equivalent, as demonstrated by Dukhin and Shilov (1974), owing to Eq. (9.160). Combining Eqs. (9.160), (9.164), and (9.165), one obtains ηsusp σ susp E∞ = ∞ = Esusp σ η

(9.166)

In the limit of an infinitely dilute suspension, φp → 0, the two conductivities become identical, and accordingly, the external and internal electric fields and mobilities will also become identical. Carrique et al. (2001) noted that the external electrophoretic mobility, η, obtained by solving the cell model with the Levine–Neale boundary condition approaches the Smoluchowski result, η = ǫζ /µ, as κa → ∞ for all particle volume fractions, φp . In contrast, the internal (or suspension) mobility, ηsusp , based on the Shilov–Zharkikh cell model, Eq. (9.163), has significantly different values for different volume fractions as κa → ∞. However, if one converts the internal mobility to the external mobility using the relation η = ηsusp

σ∞ σ susp

(9.167)

9.4 ELECTROPHORETIC MOBILITY IN CONCENTRATED SUSPENSIONS

351

which follows from Eq. (9.166), the Shilov–Zharkikh cell model provides values of the external mobility, which are identical to the Levine–Neale mobility. The conductivity ratio in Eq. (9.167) can be obtained from Eq. (9.151). Figure 9.22 depicts the difference between the internal electrophoretic mobility, U/Esusp , obtained using Carrique’s expression, Eq. (9.163), and the external mobility, U/E∞ , obtained using the Levine–Neale boundary condition, Eq. (9.148). Notably, converting the suspension mobility to the external mobility using Eq. (9.167) leads to identical results as given by Eq. (9.148). It is therefore evident that employing the Shilov–Zharkikh boundary condition we obtain the suspension mobility, which strongly varies with particle volume fraction for large κa. However, when we determine the corresponding external mobility, the results based on the Shilov–Zharkikh cell model become identical to those obtained using the Levine–Neale cell model. Both cell models predict that the external mobility approaches the Smoluchowski limit as κa → ∞. Similar consequences arising from the use of the Shilov–Zharkikh and the Levine–Neale electrophoretic mobilities have also been discussed by Ding and Keh (2001). Disregarding the implications of the two definitions of electrophoretic mobility has sometimes provided misleading conclusions regarding the validity of the Shilov–Zharkikh cell model. For instance, Kozak and Davis (1986) criticized the

Figure 9.22. Dimensionless suspension electrophoretic mobility plots obtained by Carrique et al. (2001) based on Shilov–Zharkikh cell model (dashed lines) at different particle volume fractions. When one converts these internal mobilities to external mobilities using Eq. (9.167), the solid lines are obtained. These solid lines are identical to the results obtained using the Levine–Neale cell model, see Figure 9.20. All the solid lines, obtained for different particle volume fractions, φp , approach the Helmholtz–Smoluchowski electrophoretic mobility as κa → ∞.

352

ELECTROPHORESIS

Shilov–Zharkikh model after noting that the calculated mobility based on the Shilov– Zharkikh boundary condition did not approach the Smoluchowski limit at large κa. They arrive at this conclusion due to the fact that the correct definition of the mobility was not employed while assessing the validity of the Shilov–Zharkikh cell model. However, from the foregoing results, it is clear that the Shilov–Zharkikh and Levine– Neale cell models provide identical results for the external electrophoretic mobility, η, and both models appear to be correct. 9.4.6

Accuracy of the Cell Model Predictions

In the foregoing discussion on two types of cell models (Levine–Neale and Shilov– Zharkikh), we resolved one of the two discrepancies arising from the use of different boundary conditions. The first discrepancy stems from the use of different definitions of the electric field (external and internal) in the two cell model approaches. At the end, we observed that the two types of cell models provide identical results for electrophoretic mobility once a consistent definition of electric field is used in either approach. There is a second discrepancy in the formulation of the outer cell surface boundary condition between the two approaches. This discrepancy arises from use of the gradient of the perturbed potential, ∇δψ, in the Levine–Neale boundary condition, as opposed to use of the perturbed potential, δψ, in the Shilov–Zharkikh boundary condition. This implies that one of the cell models still incorrectly specifies the outer cell surface boundary condition. Yet, once the first discrepancy between the two cell models is resolved, the predictions of the electrophoretic mobility obtained using both the cell models become identical. This leads to the question as to why, in spite of the second discrepancy between the boundary conditions, the two cell models yield identical predictions of the electrophoretic mobilities? Furthermore, one is faced with the question as to which of the two cell models use the correct boundary condition? The questions raised above have tremendous ramifications in the literature concerned with electrophoresis in concentrated dispersions, as well as sedimentation of concentrated suspensions. Broadly, two criteria are employed to test the validity of a model for electrophoretic mobility. A successful model should abide by both these criteria. These are: (i) The Smoluchowski criterion, which states that the electrophoretic mobility (more precisely, the external mobility, η) should approach the Smoluchowski limit result, η = ǫζ /µ, as κa → ∞ for every particle volume fraction, and, (ii) The Onsager reciprocity criterion, which implies that the electrophoresis and sedimentation being reciprocal processes, should lead to a specified linear relationship between the sedimentation potential and electrophoretic mobility. Violation of the Smoluchowski criterion implies that the predicted electrophoretic mobility will not become independent of φp as κa → ∞. Violation of the Onsager reciprocity criterion implies that when one uses the same cell model and perturbation technique to obtain the electrophoretic mobility and the sedimentation potential independently, the two quantities will not be related by the Onsager reciprocal relationship.

9.4 ELECTROPHORETIC MOBILITY IN CONCENTRATED SUSPENSIONS

353

The Onsager reciprocal relationship between the sedimentation potential, Esed , and the electrophoretic mobility, η, is (see Chapter 10 for details)

Esed = −

φp (ρp − ρ)η g σ∞

(9.168)

Here, φp is the particle volume fraction, ρp and ρ are the particle and fluid material densities, respectively, σ ∞ is the bulk electrolyte solution conductivity, and g is the gravitational acceleration. The electrophoretic mobility computed by both the Levine–Neale and the Shilov– Zharkikh cell models satisfy the Smoluchowski criterion. However, the Levine–Neale cell model does not provide the correct Onsager relationship between the sedimentation potential and the electrophoretic mobility (Shilov et al., 1981), thereby violating the Onsager reciprocity criterion. In other words, the sedimentation potential obtained by substituting η calculated using the Levine–Neale cell model for electrophoresis in Eq. (9.168) will not be the same as Esed obtained by directly solving the governing electrochemical transport equations for sedimentation using the Levine–Neale cell model. To satisfy the Onsager reciprocity criterion, however, both procedures should provide the same result. The Shilov–Zharkikh cell model, on the other hand, satisfies the Onsager reciprocity criteria. The Levine–Neale cell model violates the Onsager relationship due to the use of the perturbed potential gradient, ∇δψ, in defining the outer cell surface boundary condition. In other words, Eqs. (9.122) and (9.146) are inappropriate for defining the electrical conditions at the outer cell boundary. Fortunately, despite the use of this boundary condition, the Levine–Neale perturbation solution for the electrophoretic mobility turns out to be accurate. The Levine–Neale perturbation approach represents a first order perturbation expansion in zeta potential. Up to that order, the model provides correct results for the electrophoretic mobility. If higher order perturbations are included, the error in the Levine–Neale cell model would manifest itself in the calculated electrophoretic mobility. The inconsistency with the Levine–Neale boundary condition becomes more prominent when applying the cell model based perturbation approach for obtaining the sedimentation potential. As will be discussed in Chapter 10, direct application of the perturbation technique using the Levine–Neale boundary condition for the sedimentation problem will provide an incorrect dependence of the sedimentation potential on volume fraction, φp . To summarize, the Levine–Neale cell model results, as well as the model of Ohshima presented in this section fortuitously provide the correct dependence of the electrophoretic mobility on volume fraction and κa for concentrated suspensions. However, it is not advisable to apply this model directly to determine the sedimentation potential. Consequently, to obtain a correct prediction of the sedimentation potential of concentrated suspensions employing the Levine–Neale or Ohshima cell models for low zeta potential, one can first calculate the electrophoretic mobility and then use the Onsager reciprocal relationship, Eq. (9.168), to determine the sedimentation potential from the electrophoretic mobility.

354

ELECTROPHORESIS

9.5 CIRCULAR CYLINDERS NORMAL TO THE ELECTRIC FIELD Electrophoresis of a swarm of fibres (circular cylinders) is important for the study of the electrokinetic transfer phenomenon of colloidal particles in fibrous media. For the case of the electric field normal to a circular cylinder, Henry (1931) analyzed the electrophoretic mobility of an isolated cylinder. Abramson et al. (1942) further developed Henry’s analysis of an isolated cylinder to obtain results for intermediate values of κa where “a” represents the circular cylinder radius. Van der Drift et al. (1979) applied Henry’s approach, relaxing the constraint of linearization of the Boltzmann equation that describes the charge density. The extension of the analysis for the electrophoretic mobility for the case of a mat of cylinder fibers was conducted by Guzy et al. (1983), and Kozak and Davis (1986). The latter combined Henry’s approach with Kuwabara’s cell model to analyze the electrophoretic mobility of a mat of cylinders normal to the electric field strength. The governing equations are similar to those of Levine and Neale (1974a), except that they are cast in cylindrical coordinates. The Debye Hückel approximations are employed. For the case of the cell model, the fibrous mat concentration is given by φp =

 a 2 b

=1−f

(9.169)

where a is the cylinder radius, b is the outer shell radius, φp and f are the mat volume fraction and porosity, respectively. Strictly speaking, the cell model is applicable to an ordered array of fibres as shown in Figure 9.23. The effect of the mat porosity is given in Figure 9.24. As is for the case of a suspension of spheres, the effect of particle concentrations is important when the electric double layers overlap, κa ≪ 1 (Kozak and Davis, 1986). The correction factor is shown in Figure 9.24 and is given by U=

ǫζ E∞ g3 (κa, f ) µ

(9.170)

where U is the electrophoretic velocity of a cylinder within the mat.

Figure 9.23. Electrophoretic mobility of a mat of cylinders. Cell model schematic.

9.5 CIRCULAR CYLINDERS NORMAL TO THE ELECTRIC FIELD

355

Figure 9.24. Electrophoretic mobility of a mat of cylinders moving normal to their axis. Data are due to Kozak and Davis (1986), redrawn. Henry’s asymptote refers to porosity of unity (single cylinder); Smoluchowski asymptote is for large κa and a single cylinder.

For large values of κa [κa > 20 and for (κb − κa) > 4] the solution for the electrophoretic mobility is given by   ǫζ 55.7865 + 71.845f (2 + f ) 6.9984 + 11.4984f −4 η= + − + O(κa) 1− µ κaf (κa)2 f (κa)3 f (9.171) For an isolated cylinder f = 1 and Eq. (9.171), which is valid for κa > 20 becomes   ǫζ 3 18.5 127.6 η= 1− + − + · · · (9.172) µ κa (κa)2 (κa)3 The above expression for an isolated cylinder is very similar to the corresponding expression due to Henry (1931) for the case of an isolated sphere, which is given as (Hunter, 1981):   ǫζ 25 220 3 η= + − + · · · for κa ≥ 20 (9.173) 1− µ κa (κa)2 (κa)3 For the limit of small κa(κa < 0.01), the asymptotic solution for all f for the case of cylinder is given by   ǫζ 2(1 − f ) η= 1− (9.174) 2µ 2(1 − f ) − κa ln κa In the limit of f → 1 and κa ≪ 1, Eq. (9.174) reduces to Hückel’s expression for the case of an isolated cylinder with the electric field normal to the axis of the cylinder.

356

ELECTROPHORESIS

Equation (9.174) together with Figure 9.24 clearly show that when the double layer is thick, the presence of the surrounding fibers (cylinders) has a significant effect on the electrophoretic mobility of the cylinders. For κa < 0.1, the overlapped electric layers would reduce the electroosmotic velocity even in a very dilute mat of cylinders. An important observation is that the expressions derived for the electrophoretic velocity for a swarm of spheres or a mat of cylindrical fibers are equally valid for electroosmotic flow within a rigid bed of spheres or a mat of fibres. This is because both processes involve the relative notion of solid particles and electrolyte under the influence of an externally applied field, in the absence of an applied pressure gradient (Levine and Neale, 1974a).

9.6 NOMENCLATURE a A, Ac b D Er Ex Ey Eθ E∞ Esed E∞ Esusp e f (κa) f fb FE FH  F κa, K ′ g g1 (κa, φp ) g2 (κa, φp ) g3 (κa, f ) G(κa, φp ) i i∞ j∗∗ i kB

radius of a sphere or a cylinder, m cross-sectional flow area, m2 outer radius of the cell model particle diffusion coefficient, m2 /s electric field in the radial direction, V/m electric field in x-direction, V/m electric field in y-direction, V/m electric field component in the angular direction, V/m electric field magnitude far from a particle, V/m sedimentation potential, V/m electric field vector far from the particle, V/m suspension (or internal) electric field, V/m magnitude of electronic charge, C Henry’s function, dimensionless porosity, 1 − φp body force (force per unit volume) acting on fluid, N/m3 electric force on a particle, N force on a particle due to hydrodynamic drag, N generalized Henry’s function defined by Eq. (9.93) gravitational acceleration, m/s2 Möller et al. (1961) function to account for κa and spherical particle volume fraction, dimensionless Levine and Neale (1974a,b) function to account for κa and spherical particle volume fraction, dimensionless Kozak and Davis (1986) function to account for κa and cylindrical fibre mat porosity, dimensionless Ohshima’s function to account for κa and particle volume fraction suspension current density, A/m2 external current density in electrolyte solution, A/m2 flux of i th ionic species, m−2 s−1 Boltzmann constant, J/K

9.6 NOMENCLATURE

K′ M∗ ni n∞ n N0 p Pe qs qs∗ Qf Qs r t T ur ux uθ u u∞ U U∗ USt UH S vi (x, y) zi z

ratio of a particle to medium electric conductivity defined by Eq. (9.92) dimensionless mobility = 2U ∗ /β ∗ number concentration of i th ionic species, m−3 ionic number concentration in the bulk solution, m−3 unit surface normal vector number concentration of particles in suspension, m−3 pressure, Pa particle Peclet number, Ua/D surface charge density, C/m2 dimensionless surface charge density defined by Eq. (9.100) total free charge, C total surface charge, C radial coordinate, m dummy variable absolute temperature, K local radial fluid velocity, m/s local fluid velocity in x-direction, m/s local angular fluid velocity, m/s local fluid velocity vector, m/s velocity of fluid far from the particle, m/s electrophoretic particle velocity, m/s; fluid velocity far from the particle; fluid velocity at the outer surface of a cell dimensionless electrophoretic velocity defined by Eq. (9.101) Stokes free settling velocity, m/s hindered settling velocity, m/s ionic species velocity, m/s Cartesian coordinates, m valence of i th ionic species absolute value of the valency for a (z : z) electrolyte

Greek Symbols β∗ δµi δψ ǫ ǫp η ηsusp κ λi ζ ζ∗

357

dimensionless electric field strength defined by Eq. (9.99) perturbation in i th species chemical potential perturbation in electric potential, V dielectric permittivity of a material, C/mV porosity of a suspension = 1 − αp electrophoretic mobility based on external electric field, E∞ , m2V−1 s−1 electrophoretic mobility based on suspension electric field, Esusp , m2V−1 s−1 inverse Debye length, m−1 Stokes Einstein friction factor, kB T /Di zeta potential, V dimensionless zeta potential = zeζ /kB T

358

ELECTROPHORESIS

θ µ µi ρf = σ σ′ σ∞ σ susp φ φp ψ ψ eq , ψ (0) ∇ ∇2 ω

angular coordinate fluid viscosity, Pa s electrochemical potential of i th ionic species free charge density, C/m3 total hydrodynamic stress tensor, N/m2 particle electric conductivity, S/m bulk electrolyte solution conductivity, S/m suspension conductivity, S/m potential due to external electric field, V particle volume fraction in a suspension total electric potential, V potential due to stationary double layer, V del operator, m−1 Laplace operator, m−2 vorticity, s−1

9.7 PROBLEMS 9.1. Calculate the electrophoretic velocity U of a charged spherical particle of radius 0.1 µm and zeta potential 35 mV suspended in an aqueous NaCl solution at 300 K when an electric field of 105 V/m is applied. Plot the electrophoretic mobility of the particle as a function of κa for 0.1 < κa < 100. 9.2. We intend to separate the components of a very dilute aqueous binary colloidal suspension using electrophoresis. The suspension contains two types of spherical colloidal particles, one having a radius of 50 nm and a zeta potential of −30 mV, while another of radius 20 nm and unknown zeta potential. Figure 9.25 shows the separation scheme. Both particles start at the same location and traverse the length L at different times when an electric field is applied between the electrodes as shown.

Figure 9.25. Electrophoresis of two spherical colloidal particles of different size and surface potential.

9.8 REFERENCES

359

The length L is 200 µm. Under the external field, the larger particle travels the distance from start to finish in 12 seconds. What is the zeta potential of the smaller particle if it reaches the finish line in 26.5 seconds? The electrolyte (1 : 1) concentration is 0.01 M and the solvent viscosity is 0.001 Pa s. Since the suspension is very dilute, you can assume that the particle mobility can be adequately represented by the single particle mobility expressions. Use the approximate curve fit formula, Eq. (9.87) to calculate Henry’s function. 9.3. Consider the suspension conductivity expression given by Eq. (9.151). Assume that the aqueous solution is made up of a (z : z) electrolyte with equal diffusion coefficients. Plot the variation of σ susp /σ ∞ with particle volume fraction for eζ /kB T = 1.5, −1.5 and κa = 10, 20, and 100. As well, plot the corresponding variation for eζ /kB T = 0 and very large κa. 9.4. Evaluate the variation of the normalized conductivity of a suspension, given by σ susp /σ ∞ , with particle volume fraction for different aqueous systems. Assume that the charged particles have a zeta potential of 35 mV. The normalized conductivity ratio, σ susp /σ ∞ , gives a measure of the change in the electric conductivity due to the presence of the charged particles relative to the solution conductivity in the absence of particles. (a) Sodium chloride solution at 0.001 M at 25◦ C. (b) Hydrogen chloride solution at 0.001 M at 25◦ C. (c) Sodium hydroxide solution at 0.001 M at 25◦ C. (d) Calcium chloride solution at 0.001 M at 25◦ C. 9.5. Evaluate the variation of the relative conductivity ratio of a suspension, susp σ susp /σζ =0 , with volume fraction for different aqueous systems. Assume that susp the charged particles have a zeta potential of 35 mV. Here σζ =0 is defined as the conductivity of a suspension made up of particles having zero electric surface susp charge. Consequently, σ susp /σζ =0 gives a measure of the suspension electric conductivity due particles surface charge. (a) Sodium chloride solution at 0.001 M at 25◦ C. (b) Hydrogen chloride solution at 0.001 M at 25◦ C. (c) Sodium hydroxide solution at 0.001 M at 25◦ C. (d) Calcium chloride solution at 0.001 M at 25◦ C. 9.8 REFERENCES Abramson, H. A., Moyer, L. S., and Gorin, M. H., The Electrophoresis of Proteins, Reinhold, New York, (1942). Booth, F., The cataphoresis of spherical, solid non-conducting particles in a symmetrical electrolyte, Proc. Roy. Soc. Lond., 203A, 514, (1950). Carrique, F., Arroyo, F. J., and Delgado, A. V., Electrokinetics of concentrated suspensions of spherical colloidal particles: Effect of a dynamic Stern layer on electrophoresis and DC conductivity, J. Colloid Interface Sci., 243, 351–361, (2001). Datta, R., and Kotamarthi, V. R., Electrokinetic dispersion in capillary electrophoresis, AIChE J, 36, 916–925, (1990). Ding, J. M., and Keh, H. J., The electrophoretic mobility and electric conductivity of a concentrated suspension of colloidal spheres with arbitrary double-layer thickness, J. Colloid Interface Sci., 236, 180–193, (2001).

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Dukhin, S. S., and Derjaguin, B. V., Electrokinetic phenomena, in Surface and Colloid Science, vol. 7, Matijevic, E. (Ed.), Wiley, New York, (1974). Dukhin, S. S., and Semenikhin, N. M., Theory of double layer polarization and its effect on the electrokinetic and electroptical phenomena and te dielectric constants of dispersed systems, Kolloid Z., 32, 360–368, (1970). Dukhin, A. S., Shilov, V., Borkovskaya, Y., Dynamic electrophoretic mobility in concentrated dispersed systems. Cell model, Langmuir, 15, 3452–3457, (1999). Dukhin, S. S., and Shilov, V. N., Dielectric phenomena and the double layer in dispersed systems and polyelectrolytes, John Wiley and Sons, New York, (1974). Gorin, M. H., in Abramson, H. A., Gorin, M. H., and Moyer, L. S., Chem. Rev., 24, 345–366, (1939). Gur, Y., Ravina, I., and Babchin, A. J., On the electrical double layer theory. i. a numerical methods for solving a generally set Poisson–Boltzmann equation, J. Colloid Interface Sci., 64, 326–332, (1978a). Gur, Y., Ravina, I., and Babchin, A. J., On the electrical double layer theory. II. The Poisson– Boltzmann equation including hydration forces, J. Colloid Interface Sci., 64, 333–341, (1978b). Guzy, C. J., Bonano, E. J., and Davis, D. J., The analysis of flow and colloidal particle retention in fibrous porous media, J. Colloid Interface Sci., 95, 523–543, (1983). Happel, J., Viscous flow in multiparticle systems, AIChE J., 4, 197–201, (1958). Happel, J., and Brenner, H., Low Reynolds Number Hydrodynamics, Prentice Hall, Englewood Cliffs, New Jersey, (1965). Henry, D. C., The cataphoresis of suspended particles, Part 1. The equation of cataphoresis, Proc. Roy. Soc. Lond., 133A, 106–129, (1931). Hückel, E., Die kataphorese der kugel, Phys. Z., 25, 204–210, (1924). Hunter, R. J., Zeta potential in Colloid Science, Academic Press, London, (1981). Keizer, A. D. E., van der Drift, W. P. J. T., and Overbeek, J. Th. G., Electrophoresis of randomly oriented cylindrical particles, Biophys. Chem., 3, 107–108, (1975). Kozak, M. W., and Davis, E. J., Electrokinetic phenomena in fibrous porous media, J. Colloid Interface Sci., 112, 403–411, (1986). Kozak, M. W., and Davis, E. J., Electrokinetics of concentrated suspensions and porous media. I. Thin electrical double layers, J. Colloid Interface Sci., 127, 497–510, (1989a). Kozak, M. W., and Davis, E. J., Electrokinetics of concentrated suspensions and porous media. 2. Moderately thick electrical double layers, J. Colloid Interface Sci., 129, 166–174, (1989b). Kuwabara, S., The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers, J. Phys. Soc. Japan, 14, 527–532, (1959). Levine, S., and Neale, G. H., The prediction of electrokinetic phenomena within multiparticle systems. I. Electrophoresis and electroosmosis, J. Colloid Interface Sci., 47, 520–529, (1974a). Levine, S., and Neale, G. H., Electrophoretic mobility of multiparticle systems, J. Colloid Interface Sci., 49, 330–332, (1974b). Miller, N. P., and Berg, J. C., Experiments on the electrophoresis of porous aggregates, J. Colloid interface Sci., 159, 253–254, (1993).

9.8 REFERENCES

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Miller, N. P., Berg, J. C., and O’Brien, R. W., The electrophoretic mobility of a porous aggregate, J. Colloid Interface Sci., 153, 234–243, (1992). Möller, J. H. N., Van Os, G. A. J., and Overbeek, J. Th. G., Interpretations of the conductance and transference of bovine serum albumin solutions, Trans. Faraday Soc., 57, 325–337, (1961). Morrison, F. A., Electrophoresis of a particle of arbitrary shape, J. Colloid Interface Sci., 34, 210–214, (1970). O’Brien, R. W., and White, L. R., Electrophoretic mobility of a spherical colloidal particle, J. Chem. Soc. Faraday Trans., II, 74, 1607–1626, (1978). Ohshima, H., Electrophoretic mobility of spherical colloidal particles in concentrated suspensions, J. Colloid Interface Sci., 188, 481–485, (1997a). Ohshima, H., Dynamic electrophoretic mobility of spherical colloidal particles in concentrated suspensions, J. Colloid Interface Sci., 195, 137–148, (1997b). Ohshima, H., Electrical conductivity of a concentrated suspension of spherical colloidal particles, J. Colloid Interface Sci., 212, 443–448, (1999). Ohshima, H., Cell model calculation for electrokinetic phenomena in concentrated suspensions: an Onsager relation between sedimentation potential and electrophoretic mobility, Adv. Colloid Interface Sci., 88, 1–18, (2000). Overbeek, J. Th. G., Theory of the relaxation effect in electrophoresis, Kolloide Beihefte, 54, 287–364, (1943). Probstein, R. F., Physicochemical Hydrodynamics, An Introduction, 2nd ed., Wiley Interscience, New York, (2003). Pintauro, P. N., and Verbrugge, M. W., The electric-potential profile in ion-exchange membrane pores, J. Membrane Sci., 44, 197–212, (1989). Reed, L. D., and Morrison, F. A., Hydrodynamic interaction in electrophoresis, J. Colloid Interface Sci., 54, 117–33, (1976). Russel, W. B., Saville, D. A., and Schowalter, W. R., Colloidal Dispersions, Cambridge University Press, Cambridge, (1989). Saville, D. A., Electrokinetic effects with small particles, Ann. Rev. Fluid Mech., 9, 321–337, (1977). Shaw, D. J., Introduction to Colloid and Surface Chemistry, 3rd ed., Butterworths, London, (1980). Shilov, V. N., Zharkikh, N. I., and Borkovskaya, Y. B., Theory of non-equilibrium electrosurface phenomena in concentrated disperse system. 1. Application of non-equilibrium thermodynamics to cell model, Colloid J., 43, 434, (1981). Smoluchowski, M. von, Versuch einer mathematischen theorie der koagulation kinetic kolloider lösungen, Z. Phys. Chem., 93, 129, (1918). Van der Drift, W. P. J. T., Keizer, A. D. E., and Overbeek, J. Th. G., Electrophoretic mobility of a cylinder with high surface charge density, J. Colloid Interface Sci., 71, 67–78, (1979). Verbrugge, M. W., and Pintauro, P. N., Transport models for ion-exchange membranes, Modern Aspects of Electrochemistry, No. 19, Conway, B. E., Bockris, J. O’M., and White, R. E. (Eds.), Plenum, New York, (1989). Wiersema, P. H., Loeb, A. L., and Overbeek, J. Th. G., Calculation of the electrophoretic mobility of a spherical colloid particle, J. Colloid Interface Sci., 22, 78–99, (1966).

CHAPTER 10

SEDIMENTATION POTENTIAL

10.1

SEDIMENTATION OF UNCHARGED SPHERICAL PARTICLES

Sedimentation of colloidal particles in liquids is of great industrial importance. In the absence of colloidal forces, for instance, when electric double layer interactions are absent between uncharged particles, their sedimentation rate is uniquely defined by the volumetric concentration of the particles, particle and liquid densities, liquid viscosity, and particle shape and size. For the case of a single sedimenting uncharged spherical particle in an infinite medium, the sedimenting velocity is given by USt = 2a 2

(ρp − ρ)g 9µ

(10.1)

where USt is the Stokes sedimentation velocity, a is the radius of the spherical particle, ρp and ρ are the particle and liquid densities, respectively, µ is the liquid density, and g is the gravitational acceleration. Equation (10.1) is known as the Stokes settling velocity and it is strictly valid for a Newtonian fluid with the particle Reynolds number being much less than unity, i.e., Rep =

2aUSt ρ ≪1 µ

(10.2)

Equation (10.1) was derived by Stokes (1851) by solving the flow momentum equations in the creeping flow limit. In Eq. (10.1), it is assumed that gravity acts downward Electrokinetic and Colloid Transport Phenomena, by Jacob H. Masliyah and Subir Bhattacharjee Copyright © 2006 John Wiley & Sons, Inc.

363

364

SEDIMENTATION POTENTIAL

and that the density of the particle is higher than that of the liquid, thereby rendering the direction of the sedimentation or settling velocity downward. Settling of hard spheres in a suspension, i.e., non-interacting uncharged spherical particles, has been the subject of a great many studies, to name a few, Brinkman (1947), Richardson and Zaki (1954), Happel (1958), Kuwabara (1959), Happel and Brenner (1983), Masliyah (1979), Batchelor (1982), Batchelor and Wen (1982), and Wang and Wen (1998). The settling of a sphere within a suspension is normally referred to as hindered settling. The settling rate of a particle within a suspension, or simply the suspension settling rate, is a strong function of the particle concentration. Experimental data have shown that for hard spheres, the hindered settling velocity, UH S , can be approximately given by UH S = (1 − φp )4.6 (10.3) USt The velocity ratio (UH S /USt ) is a measure of the effect of the particle volume fraction, φp , on the settling rate of a sphere in a monodisperse suspension. Equation (10.3) is normally referred to as hindered settling in a closed container (batch settling) and it is attributed to Richardson and Zaki (1954). The suspension hindered settling velocity, UH S , is the velocity of the suspension interface as observed by a stationary observer as shown in Figure 10.1. Due to the difficulty in tackling the problem of sedimentation of monodisperse suspensions, several methods were developed to evaluate UH S . The most used models are those of Kuwabara’s and Happel’s cell models. The cell model involves the concept that a suspension can be divided into a number of identical cells where one sphere occupies each cell. The fluid mechanics of sedimentation of a suspension of spherical particles is thus reduced to the consideration of a single sphere with its cell bounding envelope. For example, making use of the cell model approach, Kuwabara’s cell model

Figure 10.1. Settling of particles in a suspension in a closed container.

10.2

CONCEPT OF SEDIMENTATION POTENTIAL AND VELOCITY

365

provides UHKuwabara 9 1 S = 1 − φp1/3 + φp − φp2 USt 5 5

(10.4)

where φp is the volume fraction of the spheres. It is the ratio of the volume of the spherical particle to the total volume of the suspension. Determination of the settling velocity of the spheres, USt , and the suspension settling velocity, UH S can be made by using the fluid continuity and momentum conservation equations subject to appropriate boundary conditions, as was discussed in Section 9.4.1 (Happel and Brenner, 1983).

10.2

CONCEPT OF SEDIMENTATION POTENTIAL AND VELOCITY

Sedimentation of charged spherical colloidal particles in electrolyte solutions has received much more attention in the last decade. This type of electrokinetic transport is more complex than its counterpart of uncharged spherical colloidal particles. In a suspension of charged spherical particles suspended in an electrolyte solution sedimenting under gravity, the electrical double layer surrounding each particle is distorted by the liquid flow around the particle. As discussed earlier, the deformation of the electric double layer as a result of the liquid motion is referred to as the relaxation effect. The liquid (electrolyte solution) behind each particle therefore carries an excess of counterions compared to the liquid ahead of the particle. As the total current must be zero, an induced electric field is set up such that the net current becomes zero. Consequently, when electrodes are placed near the top and bottom of the settling tube containing a suspension of sedimenting charged particles, an induced electrical potential gradient can be measured. This electric potential is referred to as the migration potential or the Dorn effect (Dorn, 1878). It is the least investigated electrokinetic phenomenon discussed so far. Figure 10.2 depicts the sedimentation of a charged spherical particle in an electrolyte solution.

Figure 10.2. Sedimentation of a charged particle in an electrolyte solution.

366

SEDIMENTATION POTENTIAL

A simple derivation of the induced sedimentation potential can be made as follows. One can think of the sedimentation potential as the converse of the streaming potential that we discussed in Chapter 8. For the case of very thin electric double layers, Eq. (8.146) provides
 Ex ǫζ = (10.5) px I =0 µσ ∞ where ζ is the zeta potential of the particles and σ ∞ is the bulk electrolyte solution electrical conductivity, Eq. (6.73), given as σ∞ =

e2  2 zi Di ni∞ kB T

(10.6)

Equation (10.5) can be re-written as ESed =

ǫζ pSed µσ ∞

(10.7)

The term ESed is the induced sedimentation electric field due to particle sedimentation and pSed denotes the “driving pressure” gradient causing sedimentation.1 Although Eq. (10.5) was derived for electroosmotic flow in a straight channel, it is assumed to be valid for an arbitrary cross section, especially for the case of non-overlapping electrical double layers and low zeta (ζ ) potentials. Let us consider a box containing Np charged spheres of radius a as depicted in Figure 10.3. The pressure gradient for the case of sedimentation within the box can

1

The driving pressure in case of streaming potential is purely hydrostatic pressure, p, while in case of sedimentation potential, it is a combination of hydrostatic and gravitational pressures. This can be seen from the Stokes equation (for creeping flow), written for streaming potential and sedimentation potential. For streaming potential, we have −∇p + µ∇ 2 u + ρf E = 0 while for sedimentation potential, we have −∇p + ρg + µ∇ 2 u + ρf E = 0

where E is the electric field. One can write g = p − ρg · r where r is the position vector. Substituting the above expression in the Stokes equation for sedimentation gives −∇g + µ∇ 2 u + ρf E = 0 Thus, although the Stokes equation for sedimentation and streaming potential appear to be identical, the interpretation of the driving pressure gradient is different in the two cases.

10.2

CONCEPT OF SEDIMENTATION POTENTIAL AND VELOCITY

367

Figure 10.3. A volume element containing sedimenting spheres.

be evaluated as follows: Total driving force: Driving pressure: Driving pressure gradient:

4 3 π a Np (ρp − ρ)g 3 4 3 π a Np (ρp − ρ)g/A 3 4 3 π a Np (ρp − ρ)g/(AL) 3

(10.8) (10.9) (10.10)

where A is the cross-sectional area of the box containing a suspension of height L. Now, the number of particles per unit volume is given by N0 = Np /(AL). Hence, the particle volume fraction, φp , is given by φp = (4/3)πa 3 N0 . Here, a is the radius of the spherical particle, g is the acceleration due to gravity, and ρp and ρ are the particle and the electrolyte solution (liquid) densities, respectively. From the definitions of N0 and φp , the expression (10.10) provides the driving pressure gradient, pSed , as pSed = (ρp − ρ)φp g (10.11) Combining Eqs. (10.7) and (10.11) leads to ESed =

ǫζ (ρp − ρ)φp g µσ ∞

(10.12)

Equation (10.12) provides a measure of the sedimentation potential due to the settling of the charged spheres. Strictly speaking, one would expect that Eq. (10.12) be valid for non-overlapping electric double layers, κa ≫ 1, and for dilute suspensions, φp ≪ 1. This expression is that stated by Smoluchowski (1903, 1921). Experimental results show that Eq. (10.12) is obeyed for Pyrex glass powder (a ≃ 50 µm) settling in a KCl solution (Booth, 1954). An important statement provided by Eq. (10.12) is that the sedimentation potential, ESed , is independent of the particle radius and that ESed → 0 as φp → 0 (a single particle).

368

SEDIMENTATION POTENTIAL

It is of interest at this stage to compare Smoluchowski’s electrophoretic mobility expression, Eq. (9.41), which is valid in the limit of κa → ∞, and Smoluchowski’s sedimentation potential, Eq. (10.12), in the limit of κa → ∞. Comparing the two expressions leads to ESed =

φp (ρp − ρ)η g σ∞

(10.13)

where the directions of the vectors ESed and g are ignored. Equation (10.13) is a statement of the Onsager relationship between an induced potential due to sedimenting charged particles and movement of charged particles under the influence of an electric field. Although Eq. (10.13) was arrived at for κa → ∞, it is found to be valid for intermediate κa values and for φp ≫ 0. The microscopic electric field due to the distortion of the electric double layer, shown in Figure 10.2, reduces the sedimentation velocity of the particles. The superposition of the individual electric fields from the particles in the suspension gives rise to the macroscopic electric sedimentation potential, which is uniform for a homogeneously dispersed suspension, Ohshima et al. (1984). Smoluchowski (1921) recognized the reduction in the sedimentation rate of particles due to the presence of the induced electric field. Smoluchowski’s expression for the reduction in the Stokes settling velocity for a single sphere due to the distorted electric double layer is given by  −1 ǫ2ζ 2 USed = 1+ 2 ∞ USt a σ µ

(10.14)

for κa ≫ 1 and equal ionic mobilities (Booth, 1954; Sengupta, 1968). For small values of the term ǫ 2 ζ 2 /(a 2 σ ∞ µ) in Eq. (10.14), one can write ǫ2ζ 2 USed =1− 2 ∞ USt a σ µ

(10.15)

Let us make some estimate as to the magnitude and effect of the induced sedimentation electric potential using two examples. EXAMPLE 10.1 Sedimentation Potential. Consider a sedimenting suspension of a volume fraction of 0.1, of spherical particles with a radius of 10 µm. The zeta potential is 50 mV. The liquid medium is an electrolyte solution of 0.01 M KCl at 20◦ C. The particle density is 2600 kg/m3 . Neglect the effect of KCl on the water density and viscosity. Evaluate the sedimentation electric potential. Data: ǫ = 80.2 × 8.854 × 10−12 C/Vm; g = 9.81 m/s2 ; µ = 1.00 × 10−3 Pa s; ρ = 998.2 kg/m3 ; and z = 1 Solution

From Table 6.1 σ = 0.122 S/m at 18◦ C. σ = 0.1408 S/m at 25◦ C.

10.2

CONCEPT OF SEDIMENTATION POTENTIAL AND VELOCITY

369

The coefficient K is given by K=

1 0.1408 − 0.122 = 0.022 0.122 25 − 18

Making use of the interpolation formula provided in Section 6.2.4: σ = σ1 [1 + K(T − T1 )] The electric conductivity at 20 ◦ C is given by σ = 0.122[1 + 0.022(20 − 18)] or σ = σ ∞ = 0.127 S/m

at 20◦ C

From definition, for a 0.01 M (1 : 1) electrolyte solution, κ = 3.3 × 108 leading to κa = (3.3 × 108 )(10 × 10−6 ) = 3300 As κa ≫ 1, Eq. (10.12) can be used. Recall, Eq. (10.12) ESed =

ǫζ (ρp − ρ)φp g µσ ∞

Using the given data in the above expression, we obtain ESed =

(80.2 × 8.854 × 10−12 ) × 0.05(2600 − 998.2) × 0.1 × 9.81 0.001 × 0.127

or ESed = 0.00044 V/m

at 20◦ C.

This is a weak electrical field strength; however, with a high impedance millivolt meter, one can measure this potential (Hunter, 1981).

EXAMPLE 10.2 Settling Velocity. Evaluate the decrease in the settling velocity of a single sphere in an unbounded medium using the data of Example 10.1. Solution

From Eq. 10.14 we have  −1 ǫ2ζ 2 USed = 1+ 2 ∞ USt a σ µ

370

SEDIMENTATION POTENTIAL

Hence,

or

or

 −1 USed (80.2 × 8.854 × 10−12 )2 (0.05)2 = 1+ USt (10 × 10−6 )2 × 0.127 × 0.001 USed = (1 + 9.9 × 10−8 )−1 USt

USed ≃ 1 − 9.9 × 10−8 USt This decrease in the Stokes settling velocity is very small under the present conditions of the example where κa is very large. It is rather curious that the electric field strength, ESed , that is induced by the settling charged particles is referred to as a “potential”. In our earlier chapters we reserved the term “potential” to indicate an electric potential measured in Volts. However, for the sedimentation of charged particles, we use the term “potential” to mean the “electric field strength” with units of Vm−1 . In doing so, we are adhering to the traditional nomenclature employed in the literature pertaining to studies on sedimentation of charged particles. 10.3

DILUTE SUSPENSIONS: OHSHIMA’S MODEL

Following Smoluchowski’s studies in early 1900s on electrophoresis and sedimentation potential, Booth (1954) provided a general theory of the sedimentation phenomenon and it was closely related to the electrophoresis theory advanced by Overbeek (1943). Without considering particle-particle interaction, Booth solved a set of electrokinetic equations using regular perturbation methods to derive formulae for the sedimentation velocity and potential as a Taylor series in the zeta potential of the charged spheres. Stigter (1980) extended Booth’s analysis to allow for higher surface potentials. In this presentation, we shall follow the approach advanced by the classical treatise of Ohshima et al. (1984), which is based on O’Brien and White (1978), to analyze the sedimentation velocity of a single charged sphere and the sedimentation potential of a dilute suspension. The momentum equation governing the flow past the sphere and the equations governing the ions surrounding the charged sphere will be utilized to develop the sedimentation velocity. The sedimentation potential for a dilute suspension will be extracted using the analysis of a single sedimenting charged sphere, Ohshima et al. (1984). 10.3.1 Fundamental Governing Equations Consider a charged rigid spherical particle having a radius a, sedimenting steadily in an electrolyte solution with a downward velocity, Used , relative to a stationary

10.3

DILUTE SUSPENSIONS: OHSHIMA’S MODEL

371

Figure 10.4. Geometry of a sedimenting spherical particle.

observer. The velocity is in the same direction as the gravitational field, g. The electrolyte solution is a Newtonian fluid having viscosity µ and density ρ. The electrolyte solution contains N ionic species of valency zi with a bulk concentration ni∞ and a diffusion coefficient Di . A spherical coordinate system (r, θ, φ) is used for the analysis. The origin of the coordinate system is fixed at the center of the spherical particle. The reference frame is taken to travel with the particle. The geometry of the sedimenting spherical particle is shown in Figure 10.4. The governing steady-state equation for low Reynolds number, where the inertial term is neglected, is given by Eq. (6.112) −∇p + µ∇ 2 u + ρg − ρf ∇ψ = 0

(10.16)

The steady-state continuity equation for the electrolyte solution having a constant density under dilute conditions, Eq. (6.113), provides ∇ ·u=0

(10.17)

The ionic flux for a dilute electrolyte solution is given by the Nernst–Planck equation, which is a statement of force balance on an ionic species, Eq. (6.45) j∗∗ i = ni vi = ni u − Di ∇ni −

zi eni Di ∇ψ kB T

(10.18)

372

SEDIMENTATION POTENTIAL

To conform with the analysis of Ohshima et al. (1984), we write the ionic velocity, vi in Eq. (10.18) as Di 1 vi = u − ∇µi = u − ∇µi (10.19) kB T λi where λi is the drag coefficient of the i th ionic species, and is identical to the term fi used in Eq. (6.39). The term µi is the electrochemical potential of the i th ionic species. The electrochemical potential is given by µi = µ(o) i + kB T ln ni + zi eψ

(10.20)

is a constant term in µi . Equation (10.19) governs the velocity of the Here, µ(o) i i th ionic species, vi as influenced by the liquid velocity u and the gradient of the electrochemical potential µi . The local electric potential ψ and the space ionic charge density, ρf , are related by Poisson’s equation, Eq. (5.7), as ǫ∇ 2 ψ = −ρf

(10.21)

The space free charge density is given by Eq. (5.9) as ρf =

N 

zi eni

(10.22)

i=1

The steady state continuity (mass conservation) of the i th ionic species in the absence of a chemical reaction is given by Eqs. (6.31) and (6.61) ∇ · (ni vi ) = 0

(10.23)

Due to axial symmetry, we need only to consider the variation in the r and θ directions. The electrolyte solution velocity u is then provided by its components, ur and uθ . The unknowns in this system of equations are ur , uθ , p, vir , viθ , ni , ψ, and USed . Consequently, the number of unknowns are 5 + 3N. The fundamental governing equations provide Eq. (10.16): Eq. (10.17): Eq. (10.19): Eq. (10.21): Eq. (10.23):

2 equations 1 equation 2N equations 1 equation N equations

This gives a total of 4 + 3N equations. The above accounting indicates that the number of unknowns exceeds the number of governing equations by one. Consequently, we will need to develop an additional governing equation. This will be done after our discussion of the boundary conditions pertaining to this problem.

10.3

DILUTE SUSPENSIONS: OHSHIMA’S MODEL

373

10.3.2 Boundary Conditions Let us assume that the shear plane of the electric double layer is assumed to coincide with the surface of the sphere, r = a. As the coordinate system is fixed on the particle center, and travels downward at a velocity USed , the velocity at the particle surface becomes zero relative to the moving coordinate frame of reference. At the sphere surface, r = a, No slip condition u=0

(ur = uθ = 0)

(10.24)

No electrolyte ions penetrate the sphere surface vi · ir = 0

(10.25)

where ir is a unit normal outward vector on the sphere’s surface, i.e., r = rir . At the slip or shear plane, one may impose the condition of constant electric potential ψa = ζ

(10.26)

or a constant surface charge density −ǫn · ∇ψ = qf One should, however, note that such specifications of the particle surface boundary conditions are an artifact of an incomplete problem formulation based on neglecting the electrical fields within the particle. For a complete solution of the problem, one requires the Poisson or Laplace equation to be written within the particles, with appropriate conditions of continuity of electric potential and discontinuity of electric displacement imposed at the particle surface. For radial positions far away from the sphere, r → ∞, u → −USed = −USed iz , the following boundary conditions hold: u · ir = −USed cos θ

(10.27-a)

u · iθ = USed sin θ

(10.27-b)

In other words, the radial velocity component is given as ur = −USed cos θ

(10.28)

and the angular velocity component is given by uθ = USed sin θ

(10.29)

where ur and uθ are the radial and angular components of the fluid velocity vector, respectively.

374

SEDIMENTATION POTENTIAL

At large distances from the sphere, the influence of the sphere is not felt, leading to ψ =0

(10.30)

ni = ni∞

(10.31)

and The boundary conditions for vi follows from Eqs. (10.18), (10.30), and (10.31), leading to vi = u at r → ∞. Although the sedimentation velocity USed appears in the boundary conditions, it is an unknown quantity. We need now to provide an additional governing equation to account for USed . The sedimentation velocity can be evaluated by recognizing that the net force acting on the spherical particle or an arbitrary volume enclosing the particle is zero. To this end, we consider a large spherical shell, S, of radius r enclosing the particle. The radius r of the surface S is taken to be large enough so that the net electric charge within S is zero (Ohshima et al., 1984). Consequently, one needs to consider the gravitational force, Fg , and hydrodynamic force, FH , where Fg + FH = 0

(10.32)

The gravity force acting on the material enclosed by the spherical shell, S, is given by 4 4 (10.33) Fg = π a 3 ρp g + π(r 3 − a 3 )ρg 3 3 The hydrodynamic force, FH , is given by  = FH = τ · n dS (10.34) S

=

The hydrodynamic stress tensor, τ has the following components: τrr = −p + 2µ

∂ur ∂r

(10.35)

and τrθ = µ



1 ∂ur ∂uθ uθ + − r ∂θ ∂r r



(10.36)

where τrr and τrθ are the normal and tangential components of the hydrodynamic stress, respectively. The sedimentation velocity, USed , is implicitly contained in the hydrodynamic stress components, τrr and τrθ . 10.3.3 Perturbation Approach The governing equations for the charged sedimenting sphere are non-linear and difficult to solve. To obtain a solution, Ohshima et al. (1984) assumed that the electric

10.3

DILUTE SUSPENSIONS: OHSHIMA’S MODEL

375

potential, ion concentrations, and the electrochemical potentials in the double layer around the spherical particle are only slightly perturbed from their symmetric equilibrium distribution due to the superimposition of the gravitational field. One can then write ni = n(0) i (r) + δni (r, θ)

(10.37)

ψ = ψ (0) (r) + δψ(r, θ)

(10.38)

µi =

µ(0) i

+ δµi (r, θ)

(10.39)

where the superscript (0) refers to the equilibrium state and δ signifies the perturbation from equilibrium. Through the definition of µi , given by Eq. (10.20), the perturbed variables, δni , δψ, and δµi are inter-related by δµi = zi eδψ + kB T δni /n0i

(10.40)

In the absence of a gravitational field, g, there is no flow, i.e., u = vi = 0, and one can obtain the classical solution for the electric field problem of a charged sphere in an electrolyte solution. Consequently, n(0) i obeys the Boltzmann equation where (0) n(0) i = ni∞ exp(−zi  )

(10.41)

and we obtain the conventional Poisson–Boltzmann equation 1 d r 2 dr

N
(0) (0)  i=1 ni∞ zi exp(−zi  ) 2 d 2 r = −κ N 2 dr i=1 ni∞ zi

where the Debye length κ −1 is given by  κ −1 =

and

ǫkB T

N

 (0) =

i=1

ni∞ zi2 e2

eψ (0) (r) kB T

(10.42)

(10.43)

(10.44)

Solution of Eq. (10.42) subject to the boundary conditions of  (0) =

eζ kB T

at r = a

(10.45)

at r → ∞

(10.46)

and  (0) → 0

provides the distribution of ψ (0) (r) and n(0) i (r) as a function of the radial position (noting that it is independent of angular position).

376

SEDIMENTATION POTENTIAL

Recognizing that the curl (∇×) of a scalar is zero and using the identity ∇ 2 u = ∇(∇ · u) − ∇ × (∇ × u)

(10.47)

the substitution of Eqs. (10.37) to (10.39) into the fundamental governing equations leads to N  µ∇ × ∇ × ∇ × u = ∇δµi × ∇n(0) (10.48) i i=1

and

∇ · ni(0) u − n(0) i ∇δµi /λi = 0

(10.49)

(∇µi ) · ir = ∇(µ0i + δµi ) · ir = 0

(10.50)

In the above derivation, the products of u, δni , δψ, and δµi are neglected. The sedimentation problem reduces to solving Eqs. (10.48) and (10.49). The boundary conditions for u are those given by Eqs. (10.24), (10.28), and (10.29). Making use of the boundary condition of Eq. (10.25) and the no-slip condition at r = a, one can write

leading to ∇(δµi ) · ir = 0

at r = a

(10.51)

At a location far from the sphere, Ohshima et al. (1984) set δψ → 0, and δni → 0. With the aforementioned conditions, Eq. (10.40) leads to δµi → 0

at r → ∞

(10.52)

Equations (10.51) and (10.52) are the boundary conditions for δµi . Ohshima et al. (1984) obtained solutions for ψ (0) , n(0) , u, and δµi , which were subsequently used to evaluate the sedimentation velocity and potential. They provided analytical expressions for both the sedimentation velocity and potential for low zeta potentials for highly dilute suspensions of charged spheres. 10.3.4 Sedimentation Velocity: Single Charged Sphere Ohshima et al. (1984), in their classical paper, provided numerical calculations for the sedimentation velocity of a single sphere that is valid for low zeta potentials. As well, for ease of computations, they also provided an approximate expression for the sedimentation velocity that is given by USed = 1 − A1 USt




eζ kB T

2

{exp(2κa)[3E4 (κa) − 5E6 (κa)]2

+ 8 exp(κa) [E3 (κa) − E5 (κa)] − exp(2κa) [4E3 (2κa) + 3E4 (2κa) − 7E8 (2κa)]} + O(ζ 3 )

(10.53)

10.3

DILUTE SUSPENSIONS: OHSHIMA’S MODEL

377

where En (x) = x n−1



x

∞

exp(−t) dt tn

(10.54)

is an exponential integral of order n, Abramowitz and Stegun (1965), and 1 A1 = 8



where mi =

N 4 i=1 zi ni∞ mi N 2 i=1 zi ni∞

2ǫkB T λi 3µzi2 e2



(10.55)

(10.56)

Here, mi is the scaled drag coefficient of the i th ionic species and λi is the drag coefficient given as kB T (10.57) λi = Di Equation (10.53) gives the ratio of the sedimentation velocity of a charged sphere to that of an uncharged sphere, i.e., Stokes sedimentation velocity given by Eq. (10.1). Figure 10.5 shows the variation of the velocity ratio, USed /USt , for a single charged sedimenting sphere with the dimensionless surface zeta potential of eζ /kB T . Both the complete numerical solution and the approximate analytical expression, Eq. (10.53) are plotted in Figure 10.5. For a very thin electric double layer, κa → ∞, as well

Figure 10.5. Ratio of sedimentation velocity to Stokes velocity, USed /USt , as a function of dimensionless zeta potential, eζ /kB T . Solid lines represent the complete solution of Ohshima et al. (1984), and the dashed lines show the approximate solution, Eq. (10.53). Physical conditions are for KCl electrolyte solution at 25◦ C.

378

SEDIMENTATION POTENTIAL

as for a very thick double layer, κa → 0, the velocity ratio is unity and the surface charge has a negligible influence on the Stokes sedimentation velocity. The effect of the surface charge on the sedimenting velocity is evident only for the intermediate κa values and at large zeta potentials. From Figure 10.5, it can be observed that the approximate analytical expression is valid for eζ /kB T ≤ 3 for the case of (1 : 1) KCl electrolyte solution. 10.3.5 Sedimentation Potential: Dilute Suspensions Ohshima et al. (1984) considered a dilute suspension made up of identical spherical particles in an electrolyte solution. The electric field arising from each charged particle is superimposed to give the macroscopic electric sedimentation potential, ESed . One can think of ESed as being a uniform electric field strength and being the average of the gradient of the potential ψ within the suspension over the suspension volume, Vsusp . Consequently, one can write ESed = −

1 Vsusp



∇ψ(r)dVsusp = − ∇ψ

(10.58)

Vsusp

The above definition of the sedimentation potential is equivalent to that of Esusp used in Eq. (9.158). Making use of Eq. (10.38) and the fact that the volume average of ∇ψ (0) (r) is zero, the sedimentation potential is given by ESed = −

1 Vsusp



∇δψ(r, θ)dVsusp

(10.59)

Vsusp

Equation (10.59) clearly illustrates that the sedimentation potential arises from the distortion of the electric double layer and it is macroscopic. Making use of the solution for δψ(r, θ) and the fact that the average current density,

i, is zero where  1

i = i(r, θ)dVsusp (10.60) Vsusp Vsusp Ohshima et al. (1984) obtained a general solution for ESed that would require a numerical procedure for its evaluation. For the special case of dilute suspensions and low zeta potentials, Ohshima et al. (1984) provided a useful approximate analytical expression for the sedimentation potential that is valid for low zeta potentials, i.e., eζ /kB T ≤ 2. ESed (κa) = ESed = −

ǫζ (ρP − ρ)φp gH (κa) + O(ζ 2 ) µσ ∞

(10.61)

where H (κa) = 1 + exp(κa) [2E5 (κa) − 5E7 (κa)]

(10.62)

10.3

DILUTE SUSPENSIONS: OHSHIMA’S MODEL

379

∗ Figure 10.6. Variation of dimensionless sedimentation potential, ESed , with dimensionless zeta potential for a dilute sedimenting suspension of spheres in a KCl solution at 25◦ C. As a result of the Onsager relationship, the curves are similar to those obtained for electrophoresis, see Figure 9.10.

The exponential integral En (κa) is given by Eq. (10.54). The expression for the sedimentation potential, Eq. (10.61), agrees with the study of Saville (1982). As stated by Ohshima et al. (1984), the negative sign on the right hand side of Eq. (10.61) implies that for positively charged spheres, the electric field acts upward in the opposite direction of gravity along which sedimentation takes place. Such a result is to be expected as the counterion charge cloud for positively charged particles is swept to the trailing edge of the particle. ∗ Figure 10.6 shows the variation of the dimensionless sedimentation potential, ESed , with the dimensionless zeta potential, eζ /kB T , obtained using numerical calculations, where 3µeσ ∞ |ESed | ∗ = ESed (10.63) 2ǫkB T (ρp − ρ)φp |g| ∗ increases with the surface For both the limiting cases of κa → ∞ and κa → 0, ESed zeta potential. As with the case of electrophoretic mobility, O’Brien and White (1978), ∗ curve exhibits a maximum at eζ /kB T ≈ 5 for κa ≥ 3. See Figure 9.10 for the ESed the corresponding behavior of the electrophoretic mobility. For low zeta potentials,

380

SEDIMENTATION POTENTIAL

we can make use of the analytical expression for ESed , Eq. (10.61), to give ∗ ESed

3 = 2




eζ kB T



H (κa)

(10.64)

Equation (10.64) is strictly valid for low zeta potentials except for the special limiting cases of κa → ∞ and κa → 0, where it is valid for higher zeta potentials. It is interesting to note that in the limit of κa → ∞, H (κa) → 1, and Eq. (10.61) degenerates to the expression given by Eq. (10.12), which was obtained using heuristic arguments. The function H (κa) is related to Henry’s function, f (κa) used in electrophoretic analysis. The relationship is given as H (κa) =

2 f (κa) 3

(10.65)

In the limits of κa → 0, H (κa) = 2/3 and f (κa) → 1, while for κa → ∞, H (κa) → 1 and f (κa) → 3/2. Recalling Henry’s analysis for the electrophoretic mobility at low zeta potentials, Eq. (9.86), 2 ǫζ η= f (κa) (10.66) 3 µ we can write the electrophoretic mobility of a single spherical particle as η(κa) =

ǫζ H (κa) µ

(10.67)

where the electrophoretic mobility is defined without any ambiguity as η(κa) = η =

U E∞

Combining Eqs. (10.61) and (10.67) leads to ESed (κa) = ESed = −

φp (ρp − ρ)η g σ∞

(10.68)

Equation (10.68) is a statement of the Onsager relationship between the sedimentation potential and the electrophoretic mobility. For the present, let us state that Eq. (10.68) is correct only for a dilute system, φp → 0. Equation (10.68) was also derived by de Groot et al. (1952) using irreversible thermodynamics. Although the derivation presented here was for the case of small values of zeta potential, where both the sedimentation potential and the electrophoretic mobility were linked together through the function, H (κa), Ohshima et al. (1984) showed that Eq. (10.68) is also valid for high values of zeta potential.

10.4

SEDIMENTATION POTENTIAL OF CONCENTRATED SUSPENSIONS

381

In conformity with the manner used to non-dimensionalize electrophoretic mobility, we can write 3U ∗ 3µe η= (10.69) η∗ (κa) = 2ǫkB T 2β ∗ where η∗ is the dimensionless electrophoretic mobility that was previously defined as 3U ∗ /2β ∗ , Section 9.3.5. From Eqs. (10.63), (10.68), and (10.69), we can write ∗ (κa) = η∗ (κa) ESed

(10.70)

The equality of Eq. (10.70) is another statement of the Onsager relationship, which is also valid for higher zeta potentials. In Figure 9.10, the dimensionless electrophoretic mobility, 3U ∗ /2β ∗ , was plotted against the scaled zeta potential, eζ /kB T , whereas ∗ , is plotted against in Figure 10.6, the dimensionless sedimentation potential, ESed eζ /kB T . Comparison between Figures 9.10 and 10.6 would indicate the reciprocity ∗ ∗ is equivalent to η∗ as indicated by Eq. (10.70). and η∗ , where ESed between ESed 10.4 SEDIMENTATION POTENTIAL OF CONCENTRATED SUSPENSIONS It was mentioned in Section 10.1 that sedimentation of uncharged monodisperse concentrated suspensions still poses theoretical challenges toward determination of the sedimentation rate. The physical problem becomes more complex when the particles are charged. In order to facilitate the analysis, many workers have resorted to the use of cell models. In Chapter 9, we showed the use of Kuwabara’s cell model by Levine and Neale (1974) and Ohshima (1997) for the study of electrophoresis of spheres in concentrated suspensions. In a similar manner, Levine et al. (1976) combined Henry’s approach with Kuwabara’s cell model to evaluate the sedimentation velocity and potential for non-overlapping double layers and low zeta potentials. To facilitate the analysis of the problem of sedimentation potential in concentrated suspensions of charged spherical particles, the Kuwabara cell model was first employed by Levine et al. (1976). In a similar manner, Ohshima (1998) carried out sedimentation potential analysis using the approach of Levine et al. (1976), where he derived an analytical expression for the sedimentation potential that is applicable for low zeta potentials and non-overlapping electric double layers of adjacent particles. The primary modification of Ohshima (1998) to the original cell model approach of Levine et al. (1976) pertains to the introduction of the total electrochemical potential of the ions, µi , and its perturbation, δµi , in presence of external forces (in this case gravity). Carrique et al. (2001) extended the study of Ohshima (1998) for higher zeta potentials. Their study utilized the Shilov–Zharkikh cell model (Shilov et al., 1981), which, according to the discussion in Chapter 9, provides a self-consistent specification of the outer cell surface boundary condition that obeys Onsager reciprocal relationship between electrophoretic mobility and sedimentation potential. The solution to the sedimentation potential problem for concentrated systems, i.e., φp ≫ 0, can be analyzed in a similar manner as to the problem of electrophoresis

382

SEDIMENTATION POTENTIAL

using a perturbation analysis. The governing continuity, momentum and electrokinetic equations are similar to those presented in Section 10.3. With the use of Kuwabara’s cell model, the outer cell boundary at r = b determines the system volume fraction, φp where φp = (a/b)3 . Consequently, the outer boundary conditions are set at r = b rather than at r → ∞ as was the case for dilute suspensions. To that end, the boundary conditions at the outer cell envelope r = b can be written as Parallel flow: ur = −USed cos θ Zero vorticity:

(10.71)

∇ ×u=0

(10.72)

(0)

dψ =0 dr Electric field at cell boundary: δµi = −zi eb n · E

(10.73)

Electrically neutral cell:

and

δni ≃ 0

(10.74)

The last boundary condition specified above, Eq. (10.74), is a source of controversy in literature. As stated here, the equation corresponds to the Shilov–Zharkikh specification, with the electric field E referring to the suspension (or internal) electric field, Esusp [cf., Eq. (9.154)]. Ohshima (1998) specified this condition as δµi = 0 for the sedimentation problem, which is analogous to setting the suspension electric field to zero. The interpretation of this outer cell surface boundary condition is subject to considerable debate (Dukhin et al., 1999). A detailed discussion on the implications of this electrochemical boundary condition in the results of Ohshima (1998) and Carrique et al. (2001) is provided in Zholkovskij et al. (2006). A direct solution of the perturbation problem for sedimentation potential was obtained by Zholkovskij et al. (2006). To provide a correct description of the perturbation problem for sedimentation potential, they used the boundary condition Eq. (10.74) based on the Shilov–Zharkikh cell model in accordance with the discussion presented by Dukhin et al. (1999). To avoid ambiguity and any confusion arising from the use of different electrochemical boundary conditions at the outer cell surface in the perturbation analysis of sedimentation potential, we will present here a different approach for calculation of the sedimentation potential. Following the discussion on the Onsager reciprocity criterion in Chapter 9 (see Section 9.4.6), it may be noted that predictions of the electrophoretic mobility can be used in the Onsager relationship between sedimentation potential and electrophoretic mobility to determine ESed . Thus, if one can determine the electrophoretic mobility of a concentrated suspension accurately, the sedimentation potential for the suspension at the same particle volume fraction can be evaluated by φp (ρp − ρ) η(κa, φp ) ESed (κa, φp ) = − g (10.75) σ∞ The term η(κa, φp ) in Eq. (10.75) refers to the external electrophoretic mobility. In Chapter 9, it was observed that all the perturbation theories (based either on the Levine–Neale or the Shilov–Zharkikh cell models) eventually led to the correct expressions for the external electrophoretic mobility over the entire range of particle volume fractions and κa. Consequently, substituting the external electrophoretic mobility obtained from the Levine–Neale or Shilov–Zharkikh cell model

10.4

SEDIMENTATION POTENTIAL OF CONCENTRATED SUSPENSIONS

383

in Eq. (10.75) will lead to the correct prediction of the sedimentation potential. This was indeed found to be the case (Carrique et al., 2001; Zholkovskij et al., 2006). The expression for the external electrophoretic mobility, η(κa, φp ), based on the Levine–Neale cell model is given by Eq. (9.128) or (9.148). There are thus two approaches for evaluating the sedimentation potential in concentrated suspensions. The first approach involves direct solution of the perturbation equations for sedimentation potential based on the Shilov–Zharkikh cell model (Zholkovskij et al., 2006). The second approach involves using the external electrophoretic mobility calculated employing either the Levine–Neale or the Shilov– Zharkikh cell models in the Onsager relationship, Eq. (10.75), to indirectly evaluate the sedimentation potential. In the following, we will compare the sedimentation potentials obtained using the direct and indirect approaches, and discuss the implications of these comparisons. However, before this comparison, we present some pertinent non-dimensionalizations required for a facile representation of the results. In conformity with our previous analysis, we can non-dimensionalize the electrophoretic mobility as η∗ (κa, φp ) =

3µe η(κa, φp ) 2ǫkB T

(10.76)

and the sedimentation potential as ∗ ESed (κa, φp ) =

3µeσ ∞ |ESed | 2ǫkB T (ρp − ρ)φp |g|

(10.77)

Combining Eqs. (10.76) and (10.77), and making use of the Onsager relationship, Eq. (10.75), leads to ∗ ESed (κa, φp ) = η∗ (κa, φp ) (10.78) In terms of the approximate analytical solution that is valid for low zeta potential, given by Eq. (9.149) and the function G(κa, φp ) of Eq. (9.150), one can express the dimensionless electrophoretic mobility as
 3 eζ (10.79) η∗ (κa, φp ) = G(κa, φp ) 2 kB T Making use of Eqs. (10.78) and (10.79), we obtain an approximate analytical ∗ expression for evaluating ESed , ∗ ESed (κa, φp )

=

∗ ESed

3 = 2




eζ kB T



G(κa, φp )

(10.80)

Of interest, for a dilute suspension, in the limit of φp → 0, comparing Eqs. (10.64), (10.65), and (10.79), one obtains G(κa, φp ) = H (κa) =

2 f (κa) 3

(10.81)

384

SEDIMENTATION POTENTIAL

where H (κa) is given by Eq. (10.62) and f (κa) is Henry’s function given by Eq. (10.65) or (9.85). In the limit of φp → 0 and κa → ∞, the various functions yield: 3 H (κa) → 1, f (κa) → , and G(κa, φp ) → 1 2 and in the limit of φp → 0 and κa → 0, one obtains H (κa) →

2 , 3

f (κa) → 1,

and

G(κa, φp ) →

2 3

The latter limit is reached only when φp is very close to zero. This asymptotic behavior is evident from Figures 9.16 and 9.20. ∗∗ ∗∗ Figure 10.7 shows the variation of ESed is with volume fraction, φp , where ESed defined as   µσ ∞ ∗∗ ESed (10.82) ESed = g(ρp − ρ)ǫζ Using Eq. (10.75) for ESed , we can obtain ∗∗ ESed =

µφp η(κa, φp ) ǫζ

(10.83)

∗∗ Figure 10.7. Variation of scaled sedimentation potential, ESed , with particle volume fraction, φp , for different values of scaled Debye length (κa). Solid lines: Direct solution of the perturbation problem for sedimentation potential using the Shilov–Zharkikh cell model (Zholkovskij et al., 2006). Dashed lines: Predictions obtained by substituting the electrophoretic mobility in the Onsager reciprocal relationship. Equations (10.84) and (9.150) (Ohshima, 1997) were employed for obtaining the sedimentation potential using this indirect approach.

10.4

SEDIMENTATION POTENTIAL OF CONCENTRATED SUSPENSIONS

385

where η(κa, φp ) is the external electrophoretic mobility. Here, we use Eq. (9.149) to obtain the electrophoretic mobility. Substitution of Eq. (9.149) in (10.83) yields ∗∗ ESed = φp G(κa, φp )

(10.84)

where the function G(κa, φp ) is given by Eq. (9.150). The solid lines in Figure 10.7 are obtained using perturbation analysis for the sedimentation potential given by Zholkovskij et al. (2006). The dashed lines are obtained by substituting Ohshima’s approximate expression for electrophoretic mobility in terms of G(κa, φp ), given by Eq. (9.150), in Eq. (10.84). The two results are virtually indistinguishable, which is a clear demonstration of two facts, namely (i) the Shilov–Zharkikh cell model conforms to the Onsager reciprocity criterion, and (ii) the electrophoretic mobility determined using the Levine–Neale cell model can be used in the Onsager relationship, Eq. (10.75), to provide an accurate estimate of the ∗∗ sedimentation potential. Figure 10.7 also shows that ESed exhibits a maximum when ∗∗ , shifts plotted against φp . The maximum in the scaled sedimentation potential, ESed to higher values of φp with increasing κa. For a given particle volume fraction, φp , higher scaled sedimentation potentials are observed at higher values of κa. ∗∗ with Figure 10.8 shows the variation of the scaled sedimentation potential, ESed κa for different values of particle volume fraction, φp . The solid lines are from the

∗∗ Figure 10.8. Variation of scaled sedimentation potential, ESed , with scaled Debye length (κa) for different particle volume fractions. Solid lines: Direct solution of the perturbation problem for sedimentation potential using the Shilov–Zharkikh cell model (Zholkovskij et al., 2006). Dashed lines: Predictions obtained by substituting the electrophoretic mobility in the Onsager reciprocal relationship. Equations (10.84) and (9.150) (Ohshima, 1997) were used for obtaining this indirect estimate of the sedimentation potential. The horizontal dashed lines represent the ∗∗ limiting values of the sedimentation potential as κa → ∞. It is evident that ESed → φp as κa → ∞.

386

SEDIMENTATION POTENTIAL

direct solution of the perturbation problem for sedimentation potential according to Zholkovskij et al. (2006). The dashed lines are obtained using the indirect approach based on the Onsager reciprocal relationship [Eqs. (10.84) and (9.150)]. Once again, the agreement between the two procedures of evaluating sedimentation potential is ∗∗ is φp . This can be excellent. For large κa, the asymptotic value reached by ESed clearly observed from Eq. (10.84). Here, G(κa, φp ) approaches unity as κa → ∞ and ∗∗ → φp as κa → ∞. For the case of a dilute suspension, φp → 0, consequently, ESed ∗∗ → φp as κa → ∞ could have been easily reached from the the conclusion of ESed dilute suspension analysis. One can easily show that ∗ ESed 3 ∗∗ = E (eζ /kB T ) 2φp Sed

(10.85)

∗ /(eζ /kB T ) For the case of a dilute suspension for κa → ∞ one can observe that ESed → 3/2 as κa → ∞. The plot of Figure 10.8 would indicate that the asymptotic ∗∗ behavior of ESed → φp as κa → ∞ is equally valid for non-dilute suspensions. One should note that Eq. (9.150) represents an approximate analytic solution for the electrophoretic mobility function, G(κa, φp ) (Ohshima, 1997). Despite the “approximate” nature of this expression, the virtual overlap between the solid and dashed lines in Figures 10.7 and 10.8 is remarkable. In fact, if one uses Eq. (9.128) or Eq. (9.148) for the electrophoretic mobility in Eq. (10.83) to obtain the dashed lines, then the solid and dashed lines in these figures will be identical. From the foregoing discussion, it is evident that analysis of sedimentation potential in concentrated suspensions is non-trivial and still poses significant difficulties. Such difficulties are owing to non-availability of direct means of measuring the sedimentation potential, as well as controversy related to application of proper boundary conditions in the perturbation analysis. One can, however, evaluate the sedimentation potential in concentrated suspensions quite accurately by employing an accurate expression for the electrophoretic mobility in the Onsager reciprocal relationship. The electrokinetic phenomena dealing with sedimentation velocity and potential still pose considerable challenges in analysis of multi-species systems having particles of different surface potentials, densities, and sizes.

10.5 a A Di e Esusp ESed Eθ Fg Fh g

NOMENCLATURE radius of a spherical particle, m cross section area, m2 diffusion coefficient of i th ionic species, m2 /s elementary charge, C electric field strength measured within the suspension, V/m induced electric field referred to as sedimentation potential, V/m electric field strength in the angular direction, V/m force on a particle due to gravity, N force on a particle due to hydrodynamic drag, N acceleration due to gravity, m/s2

10.5

g ir I kB ni ni∞ N Np N0 p pSed r r Rep T ur uθ u U USed USt UH S vi vir viθ zi

NOMENCLATURE

387

magnitude of acceleration due to gravity, |g|, m/s2 unit normal outward vector on the sphere surface total current, A Boltzmann constant, J/K ionic number concentration of the i th species, m−3 bulk ionic number concentration of the i th species, m−3 total number of ionic species total number of charged particles number of charged particles per unit volume, m−3 pressure, Pa induced sedimentation pressure gradient, Pa/m radial coordinate (in spherical coordinate system), m position vector, m particle Reynolds number absolute temperature, K local radial fluid velocity, m/s local angular fluid velocity, m/s local fluid velocity, m/s electrophoretic particle velocity, m/s; fluid velocity far from the particle or fluid velocity at the outer shell of a cell sedimentation velocity of a single charged particle or a suspension of charged particles, m/s Stokes settling velocity, m/s Hindered settling velocity, m/s velocity of the i th ionic species, m/s radial velocity component of the i th ionic species, m/s angular velocity component of the i th ionic species, m/s valency of the i th ionic species

Greek Symbols δ ǫ λi ζ ζ∗ ηel σ∞ θ φp κ µ µi µoi ρf

symbol for a perturbation variable electric permittivity of continuous phase electrolyte solution, C V−1 m−1 drag coefficient of i th ionic species, Js/m2 zeta potential, V dimensionless zeta potential = zeζ /kB T electrophoretic mobility of a charged particle defined as U/E∞ , m2 /Vs electric conductivity of continuous phase electrolyte solution, S/m angular coordinate particle volume fraction inverse Debye length, m−1 fluid viscosity, Pa s electrochemical potential of i th ionic species, J constant term in µi ionic free charge density, C/m3

388

SEDIMENTATION POTENTIAL

continuous medium density, kg/m3 particle density, kg/m3 normal stress acting on the r-plane in r-direction, N/m2 shear stress acting on the r-plane in θ-direction, N/m2 angular and azimuthal coordinate directions in spherical coordinates equilibrium (unperturbed) electric potential in absence of flow, V scaled equilibrium electric potential, eψ (0) /kB T mobility of i th ionic species, m2 /J s

ρ ρp τrr τrθ θ, φ ψ (0)  (0) ωi

10.6

PROBLEMS

10.1. For the case of concentrated suspensions with particles having low zeta potentials, show that the electric potential in the absence of flow, for the sedimentation potential problem, using Ohshima perturbation approach, becomes 1 d 2 (0) (r ψ ) = κ 2 ψ (0) r 2 dr subject to the boundary conditions ψ (0) = ζ sphere surface and dψ (0) =0 dr

at r = b (cell outer boundary).

(a) Show that the solution of the governing differential equation subject to the two boundary conditions is given by ψ

(0)

(r) = ζ

a   κb cosh[κ(b − r)] − sinh[κ(b − r)]  r

κb cosh[κ(b − a)] − sinh[κ(b − a)]

(b) Show that 

∇ψ (0) (r)dV = 0 V

where dV is an element in the spherical coordinate system. (c) What conclusion can you draw from the result of part (b)? 10.2. Evaluate the sedimentation velocity ratio USed /USt for sedimenting spheres of 1.2 µm radius with a density of 2600 kg/m3 in a very dilute aqueous suspension containing 10−5 M HCl at 25◦ C. The zeta potential of the spheres is 80 mV. 10.3. A suspension of charged spherical particles is allowed to settle in a 0.01 M KCl solution at 20◦ C in a graduated cylinder. The particles have a radius of 10 µm,

10.7

REFERENCES

389

a zeta potential of 35 mV and a density of 2600 kg/m3 . Assume a solution density of 998 kg/m3 and a viscosity of 0.001 Pa s. (a) Evaluate the sedimentation potential for such a settling suspension as a function of volume fraction. (b) How would the result change for particles having a radius of 2 µm in 10−3 M KCL solution?

10.7

REFERENCES

Abramowitz, M., and Stegun, I. A., Handbook of Mathematical Functions, US Dept. of Commerce, (1970). Batchelor, G. K., Sedimentation in a dilute polydisperse system of interacting spheres, Part I: General theory, J. Fluid Mech., 124, 379, (1982). Batchelor, G. K., and Wen, C. S., Sedimentation in a dilute polydisperse system of interacting spheres, Part II: Numerical results, J. Fluid Mech., 124, 495, (1982). Booth, F., Sedimentation potential and velocity of solid spherical particles, J. Chem. Phys., 22, 1956–1968, (1954). Brinkman, H. C., A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appl. Sci. Res., A1, 27–34, (1947). Carrique, F., Arroyo, F. J., and Delgado, A. V., Sedimentation velocity and potential in a concentrated colloidal suspension: Effect of a dynamic Stern layer, Colloids Surf. A, 195, 157–169, (2001). de Groot, R., Mazur, P., and Overbeek, J. Th. G., Nonequilibrium thermodynamics of the sedimentation potential and electrophoresis, J. Chem. Phys., 20, 1825–1829, (1952). Dorn, E., Ann. Physik, 3, 20, (1878). Dukhin, A. S., Shilov, V., and Borkovskaya, Y., Dynamic electrophoretic mobility in concentrated dispersed systems. Cell model Langmuir, 15, 3452–3457, (1999). Happel, J., and Brenner, H., Low Reynolds Number Hydrodynamics, Martinus Nijhoff, The Hague, (1983). Happel, J., Viscous flow in multiparticle systems: Slow motion of fluids relative to beds of spherical particles, AIChE J., 4, 197–201, (1958). Hunter, R. J., Zeta Potential in Colloid Science, Academic Press, London, (1981). Kuwabara, S., The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers, J. Phys. Soc. Japan, 14, 527–532, (1959). Levine, S., and Neale, G. H., The prediction of electrokinetic phenomena within multiparticle systems. I. Electrophoresis and electroosmosis, J. Colloid Interface Sci., 47, 520–529, (1974). Levine, S., Neale, G. H., and Epstein, N., The prediction of electrokinetic phenomena within multiparticle systems. II. Sedimentation potential, J. Colloid Interface Sci., 57, 424–437, (1976). Masliyah, J. H., Settling of multi-species particle system, Chem. Eng. Sci., 34, 1166–1168, (1979).

390

SEDIMENTATION POTENTIAL

O’Brien, R. W., and White, L. R., Electrophoretic mobility of a spherical colloidal particle, J. Chem. Soc. Faraday Trans. II, 74, 1607–1626, (1978). Ohshima, H., Healy, T. W., and White, L. R., Sedimentation velocity and potential in a dilute suspension of charged spherical particles, J. Chem. Soc. Faraday Trans. II, 80, 1299–1317, (1984). Ohshima, H., Electrophoretic mobility of spherical colloidal particles in concentrated suspensions, J. Colloid Interface Sci., 188, 481–485, (1997). Ohshima, H., Sedimentation potential in a concentrated suspension of spherical colloidal particles, J. Colloid Interface Sci., 208, 295–301, (1998). Overbeek, J. Th. G., Theorie der electrophorese, Kolloidchem. Beih., 54, 287–364, (1943). Richardson, J. F., and Zaki, W. N., Sedimentation and fluidization. Part I, Trans. Inst. Chem. Eng., 32, 35–53, (1954). Saville, D. A., The sedimentation potential in a dilute suspension, Adv. Colloid Interf. Sci., 16, 267–279, (1982). Sengupta, M., The sedimentation of non-conducting solid spherical particles, J. Colloid Interface Sci., 26, 240–243, (1968). Shilov, V. N., Zharkikh, N. I., and Borkovskaya, Y. B., Theory of non-equilibrium electrosurface phenomena in concentrated disperse system. 1. Application of non-equilibrium thermodynamics to cell model, Colloid J., 43, 434, (1981). Smoluchowski, M. von, Contribution a la theorie de l’endosmose electrique et de quelques phenomenes correlatifs, Bull. International de l’Academie des Sciences de Cracovie, 8, 182–200, (1903). Smoluchowski, M. von, in Handbuch der Electrizitat und des Magnetismus, (Graetz), 11, 336, Barth, Leipzig, (1921). Stigter, D., Sedimentation of highly charged colloidal spheres, J. Phys. Chem., 84, 2758–2762, (1980). Stokes, G. G., On the effect of the internal friction of fluids on the motion of pendulums, Trans. Cambridge Phil. Soc., 9, 8–106, (1851). Wang, H., and Wen, C. S., Interparticle potential and sedimentation of monodisperse colloid system, AIChE J., 44, 2520, (1998). Zholkovskij, E. K., Masliyah, J. H., Shilov, V. N., and Bhattacharjee, S., Notes on electrokinetic cell model approach (submitted), (2006).

CHAPTER 11

LONDON–VAN DER WAALS FORCES AND THE DLVO THEORY

11.1

DISPERSION FORCES BETWEEN BODIES IN VACUUM

So far, we have solely considered the presence of electrostatic interactions between colloidal entities. For charged entities in vacuum, the interaction was expressed in terms of the Coulomb potential, while in a dielectric medium containing free charges, the interaction between charged particles was described in terms of a screened electric double layer potential. All the electrokinetic phenomena described so far considered only one type of interparticle force, which was electrostatic in origin. Forces of a more fundamental and more ubiquitous nature exist between every colloidal entity. Such forces between colloidal entities are always attractive in vacuum. Microscopic observations of colloidal particles reveal their tendency to form persistent aggregates induced by Brownian motion, clearly indicating the presence of an attractive force. Consequently, to understand the behavior of charged colloidal particles in a suspension, and colloidal transport phenomena, one needs to consider these forces in addition to the electrostatic forces. The mutual attraction of particles in vacuum is a consequence of dispersion forces, often called the London–van der Waals forces. The dispersion attraction forces are due to the spontaneous fluctuation of the electronic cloud in one material causing a corresponding fluctuation in neighboring material, leading, on the average, to an attractive force (Ross and Morrison, 1988; Morrison and Ross, 2002). These forces are strictly quantum mechanical in nature and they are expressed in terms of the

Electrokinetic and Colloid Transport Phenomena, by Jacob H. Masliyah and Subir Bhattacharjee Copyright © 2006 John Wiley & Sons, Inc.

391

392

LONDON–VAN DER WAALS FORCES AND THE DLVO THEORY

same oscillator strengths as appear in the equations for the dispersion of light or electromagnetic waves. When we consider monatomic molecules (such as noble gases, e.g., Helium and Argon), a fair idea about the origin of the dispersion forces can be obtained by considering the atoms to have a positively charged nucleus and a negatively charged electron cloud, which is in a constant state of oscillation around the nucleus. Such a system can behave as an electromagnetic dipole. When another atom of the noble gas experiences the influence of such a dipole, the oscillations of its electron cloud are also affected, and so is its dipolar nature. Fluctuating dipoles can also be induced in a non-polar atom or molecule by absorption of photons from the background electromagnetic radiation field. Interaction between such a pair of atomic dipoles gives rise to the attractive dispersion forces. When the atoms are brought close together, such that their electron clouds overlap, they experience a strong repulsive force. The classical Lennard–Jones potential summarizes the interaction between such a pair of atoms through an empirical equation of the form uLJ = ǫD


  σ 12  σ 6 − r r

(11.1)

where ǫD is a characteristic energy of the dipolar interaction, and σ is the distance of neutral approach, both being empirical constants. The second term in Eq. (11.1) represents the dependence of the attractive dispersion force on the separation distance r between two atoms. The attractive force with the inverse r 6 dependence arises due to several types of dipolar interactions, namely, London interactions, dipole-dipole (Keesom) interactions, dipole-induced dipole (Debye) interactions, etc. (Israelachvili, 1991). The Lennard–Jones potential forms the basis for the calculation of non-bonded interactions in most atomic calculations like molecular mechanics simulations. These interactions are considered to be pairwise additive, and can be employed to determine interaction between polyatomic molecules. The situation becomes somewhat more complex when we consider colloidal particles of much larger dimensions comprising millions of atoms. Calculation of the dispersion interactions between such large particles is primarily achieved through two techniques: One is based on a microscopic molecular model, which is attributed to Hamaker (1937) and the other is based on a macroscopic continuum model of condensed media, attributed to Lifshitz (1956). In the microscopic model, the attraction between particles is calculated by summing the attractive energies between all pairs of atoms in the separate particles, ignoring multibody perturbations. This approximation is equivalent to predicting the spectra of condensed media as the sum of the molecular spectra. Corrections have been introduced to account for such factors as third body perturbations, the effect of intervening material, and retardation of the dispersion forces due to the finite speed of light. In the macroscopic model of particle-particle dispersion attractions, the lowering of the zero-point energy of a particle, due to the coordinator of its instantaneous electric moments with those of a nearby particle, is calculated by quantum electrodynamics

11.2

HAMAKER’S APPROACH

393

(Ross and Morrison, 1988). The expressions derived require as data the dielectric susceptibilities of the particles as a function of frequency and are more complex than those from the Hamaker theory. The complexity of Lifshitz formulae and the difficulty of obtaining the necessary material constants have hampered its use. Ninham and Parsegian (1970) have, however, developed a numerical method to approximate the necessary material functions from a few, readily obtained values, and have made the use of this theory applicable for some common materials. In the Hamaker model, the free energy of interaction separates into a materialdependent constant (called the Hamaker constant) and a geometry-dependent integral. The Lifshitz theory, on the other hand, gives such a separation of terms only for the special case of interactions between parallel plates. In this chapter we will primarily restrict our discussion of the van der Waals interactions within the context of Hamaker’s microscopic approach. Readers interested in the electrodynamic approach are referred to the original work of Lifshitz (1956), the subsequent developments made by Mahanty and Ninham (1976), and a more recent summary of the theory given by Russel et al. (1989).

11.2

HAMAKER’S APPROACH

A brief description of Hamaker’s approach is given here. The reader is referred to Russel et al. (1989), Morrison and Ross (2002), and Hunter (1991) for details and for further references. The approach of Hamaker (1937) is based on the assumption that the dispersion potential between two colloidal particles can be represented as summation of the dispersion interaction between pairs of atoms located within the two particles. The procedure starts by considering the attractive Lennard–Jones potential [cf., Eq. (11.1)] uLJ,att = −

c r6

(11.2)

where c is a constant characterizing the strength of the dispersion attraction between the atoms (c = ǫD σ 6 ). Denoting the two interacting colloidal particles by 1 and 2, and assuming the density of atoms (atoms per unit volume) within these particles to be ρ1 and ρ2 , respectively, the total interaction energy, dU12 , between all the atoms in a volume element dV1 in particle 1 and a volume element dV2 in particle 2 is dU12 = −

cρ1 ρ2 dV1 dV2 r6

(11.3)

where r is the separation between the volume elements dV1 and dV2 . The situation is depicted in Figure 11.1. Integrating Eq. (11.3) over all the volume elements in the two particles provides the dispersion energy of attraction between them. The work done by the attractive forces in bringing the two particles from infinity to a given separation distance is the

394

LONDON–VAN DER WAALS FORCES AND THE DLVO THEORY

Figure 11.1. Schematic representation of Hamaker’s approach applied to two colloidal particles.

potential due to the dispersion interaction. At a constant temperature, we obtain U = −c

  V1

V2

ρ1 ρ2 dV1 dV2 r6

(11.4)

where U is the total interaction energy between the volumes V1 and V2 of the particles 1 and 2, respectively. Hamaker’s approach is based on the following assumptions (Ross and Morrison, 1988): 1. The interactions can be considered pairwise; i.e., many body interactions are ignored. 2. The bodies are assumed to have uniform density up to the interfaces. 3. The interactions are instantaneous. 4. The intervening medium is a vacuum. 5. All the dispersion force attractions are due to one dominant frequency. 6. Effects of free charge and permanent dipoles are negligible. 7. The bodies (particles) are not distorted by the attractive forces. As a consequence of these assumptions, the term cρ1 ρ2 is a constant. We now illustrate the application of this approach to obtain the dispersion interaction energy per unit area between two infinite planar slabs (half-spaces) separated by a distance h. Consider the geometry shown in Figure 11.2. The slabs are represented in cylindrical coordinates, with the r coordinate extending to infinity. The z-coordinate starting from the origin of slab 1 (denoted by O) extends to +∞ [Figure 11.2(a)]. The z′ -coordinate of the infinite slab 2 with origin at O ′ extends to −∞ [Figure 11.2(b)]. For this geometry, the interaction energy between slab 1 and an arbitrarily chosen point P in slab 2 is first evaluated. In cylindrical coordinates, the volume element dV1 in slab 1 is represented by dV1 = 2π rdrdz

(11.5)

11.2

HAMAKER’S APPROACH

395

Figure 11.2. Evaluation of van der Waals interaction energy per unit area between two infinite flat slabs.

Denoting the distance of this volume element from point P on slab 2 by d [Figure 11.2(a)], we can write  d = r 2 + (z + H )2 (11.6)

where H is the distance of the point P from the surface of slab 1. The total interaction between point P and volume element dV1 is then given by du1P = −

cρ1 2π cρ1 dV1 = − 2 rdrdz 6 d [r + (z + H )2 ]3

(11.7)

396

LONDON–VAN DER WAALS FORCES AND THE DLVO THEORY

Integrating over the volume of the slab 1, this yields for the total interaction energy between the slab 1 and point P as u1P = −2π cρ1



∞

z=0

∞



r=0

[r 2

rdrdz + (z + H )2 ]3

(11.8)

Evaluation of the two nested integrals provides u1P = −

π cρ1 6H 3

(11.9)

The term u1P denotes the interaction energy between slab 1 and the point P located in slab 2. To evaluate the total interaction energy between the two infinite slabs, we now need to integrate the interaction energy between slab 1 and point P given by Eq. (11.9) over every point in slab 2. Toward this, let us first consider the point P to be in a differential volume element dV2 in slab 2 as shown in Figure 11.2(b). This volume element is a solid cylindrical disc of cross sectional area A and thickness dz′ . Note that since slab 1 is infinite, the interaction energy between any point located inside the volume element dV2 at a fixed z′ will by given by Eq. (11.9). In other words, the differential energy between slab 1 and the volume element dV2 is dU12 = ρ2 u1P dV2 = −

π cρ1 ρ2 Adz′ 6H 3

(11.10)

The total interaction energy per unit area of slab 2 is now given by integrating Eq. (11.10) over the entire thickness of slab 2, yielding, U12 π = − cρ1 ρ2 A 6



∞

z′ =0

1 dz′ (h + z′ )3

(11.11)

where U12 /A is the interaction energy per unit area between the two slabs, and the distance H in Eq. (11.10) is represented by H = h + z′

(11.12)

U12 A12 =− A 12π h2

(11.13)

A12 = π 2 cρ1 ρ2

(11.14)

Evaluation of the integral provides

where

is defined as the Hamaker constant.

11.2

HAMAKER’S APPROACH

397

The above illustration demonstrates how evaluation of the dispersion interactions is separated into two components. The Hamaker constant accounts for all the material properties providing the magnitude of the dispersion interactions, while the geometrical dependence is embedded in the various nested integrals that were explicitly evaluated. A similar approach can be adopted to evaluate the interaction energy for colloidal entities of various shapes. For example, for the case of planar parallel slabs of thickness δ1 and δ2 separated by a distance h in vacuum, one has   
A12 U12 1 1 1 1 =− + − − A 12π h2 (h + δ1 + δ2 )2 (h + δ1 )2 (h + δ2 )2

(11.15)

where A12 is the Hamaker constant for bodies 1 and 2. For h ≪ δ1 , δ2 , Eq. (11.15) becomes identical to Eq. (11.13). The interaction energy per unit area in Eqs. (11.13) and (11.15) has units of J/m2 . The negative sign in these expressions signifies an attractive force. The Hamaker constant generally has values in the range of 10−21 – 10−19 J. Hamaker’s approach can be employed to obtain the van der Waals interaction energy between particles of various geometries. Some pertinent expressions of van der Waals interaction energy between particles of common geometrical shapes is given in Table 11.1. Note that we use the term AH to denote the effective Hamaker constant in the expressions provided in this and subsequent tables. The effective Hamaker constant can assume different values depending on the combination of interacting materials and the intervening medium. The methodologies for calculating the effective Hamaker constant will be described later. Expressions for various geometries can be found in Mahanty and Ninham (1976) and Russel et al. (1989). One should note that the Hamaker expressions are based on exact integration of the attractive Lennard– Jones potential over two bodies, and are deemed more accurate than the corresponding estimates based on different types of approximations like Derjaguin approximation, which will be discussed later. A typical problem associated with the Hamaker expressions is that the interaction energy diverges (goes to −∞) as the surfaces of the bodies come in contact (h → 0). This unphysical divergence of the van der Waals energy stems from the fact that the repulsive (r 12 ) part of the Lennard–Jones potential, Eq. (11.1), is not considered during the integration of the atom-atom interaction energy. In reality, the strong r 12 repulsion, termed as the Born repulsion, which arises due to overlap of the electron clouds of atoms, renders the interaction between two particles strongly repulsive at finite separations ranging from 0.1 to 0.3 nm. Considering the difficulty in integrating the repulsive Lennard–Jones potential over two macrobodies, one simply accounts for this repulsion in the Hamaker expressions by employing a cut-off distance of about 0.16 nm as the contact separation between two surfaces. This minimum cutoff distance prevents the divergence of van der Waals interaction between objects of colloidal dimensions. A notable feature of the van der Waals interactions is that integrating the extremely short range attractive Lennard–Jones interactions with a 1/r 6 decay behavior over two particle volumes changes the distance dependence of these interactions dramatically.

398

LONDON–VAN DER WAALS FORCES AND THE DLVO THEORY

TABLE 11.1. Expressions for Unretarded van der Waals Interaction Energy (or Energy Per Unit Area) Between Bodies of Common Geometrical Shapes Obtained from Hamaker’s Approach. AH is the Hamaker Constant of the System and h is the Distance of Closest Approach Between the Bodies.
Two spheres of radius a1 and a2 , J 2a1 a2 AH − 2 6 h + 2a1 h + 2a2 h

2a1 a2 h2 + 2a1 h + 2a2 h + 4a1 a2   h2 + 2a1 h + 2a2 h + ln h2 + 2a1 h + 2a2 h + 4a1 a2
2a 2 2a 2 AH + 2 − 2 6 h + 4ah h + 4ah + 4a 2   h2 + 4ah + ln h2 + 4ah + 4a 2 
 AH a h a − + + ln 6 h h + 2a h + 2a +

Two equal spheres of radius a, J

Sphere of radius a and an infinite flat plate, J Sphere of radius a on the axis of a straight cylindrical capillary of radius b, J Two infinite plates of equal thickness δ, J/m2 Two semi-infinite parallel plates (half-spaces), J/m2

−

−

AH 12π




a3 4AH 3 (b2 − a 2 )3/2

1 1 2 + − h2 (h + 2δ)2 (h + δ)2 −



AH 12π h2

For instance, the interaction energy per unit area between two infinite planar slabs (or half-spaces) decays as the inverse square of the separation between the slabs. The macroscopic van der Waals interactions are thus considerably long-range, and are felt between particles that are separated by even several hundreds of nanometers. 11.2.1 Approximate Expressions for van der Waals Interaction Evaluating two volume integrals is relatively straightforward for simple geometries like two planar surfaces or two spheres, where utilization of symmetry allows considerable reduction of the computational burden involved in the integrations, even enabling analytic evaluation of the integrals. For complicated geometries, application of Hamaker’s approach is computationally burdensome, and approximate techniques are often employed to evaluate the interaction energies. Here we will discuss a classical technique, namely the Derjaguin approximation, and also briefly present a modified Derjaguin type technique called surface element integration (SEI) for approximately evaluating the van der Waals interaction energy with very little computational effort.

11.2

HAMAKER’S APPROACH

399

11.2.1.1 Derjaguin Approximation The essential principle of Derjaguin’s technique as generalized by White (1983), which was applied for evaluation of electric double layer interactions between curved geometries in Chapter 5, remains the same when one applies it for calculation of van der Waals energy. Under this approximation, one integrates the interaction energy per unit area between two infinite planar surfaces (half-spaces) over the surfaces of the interacting particles through an expression of the form (cf., Chapter 5) 2π U (h) = √ λ1 λ2

∞



h

Up (H ) dH A

(11.16)

where U is the interaction energy between two curved surfaces separated by h (distance of closest approach between their surfaces), Up /A is the interaction energy per unit area between two infinite planar surfaces at a separation H , and λ1 λ2 =



1 1 + ′ R1 R1



1 1 + ′ R2 R2



2

+ sin ϕ



1 1 − R1 R2



1 1 − ′ R1′ R2



(11.17)

Here, R1 and R2 are principal radii of curvature of body 1, R1′ and R2′ are the principal radii of curvature of body 2, and ϕ is the angle between the principal axes of bodies 1 and 2. For van der Waals interaction, the interaction energy per unit area based on Hamaker’s approach is given by Eq. (11.13). Using this expression in Eq. (11.16), one can obtain the interaction energy between two particles as AH U (h) = − √ 6 λ1 λ2



∞ h

1 AH 1 dH = − √ 2 H 6 λ1 λ2 h

(11.18)

It should be noted that we substituted A12 in Eq. (11.13) by the effective Hamaker constant, AH , to write Eq. (11.18). Introducing the principal radii of curvature for the geometry under consideration in Eq. (11.17) yields an expression for the geometrical factor, which, when substituted in Eq. (11.18) provides a simple approximate form for the van der Waals interaction. For two interacting spheres of radius a, all the principal radii of curvatures equal a, and the geometrical factor is 4 λ 1 λ2 = 2 (11.19) a Hence, Eq. (11.18) gives U (h) = −

AH a 12h

(11.20)

For a sphere and an infinite flat plate, λ1 λ2 = 1/a 2 , which yields U (h) = −

AH a 6h

(11.21)

400

LONDON–VAN DER WAALS FORCES AND THE DLVO THEORY

TABLE 11.2. Expressions for Unretarded van der Waals Interaction Energy Between Bodies of Common Geometrical Shapes Obtained using Derjaguin’s Approximation. AH is the Hamaker Constant of the System and h is the Distance of Closest Approach Between the Bodies.   Two spheres of radius a1 and a2 a1 a2 1 AH − 6 a1 + a2 h Two equal spheres of radius a

−

Sphere of radius a and an infinite flat plate

−

AH a 12h

AH a 6h √ AH a1 a2 − 6h

Two crossed cylinders of radius a1 and a2 Two parallel cylinders of length L and radii a1 and a2

−

AH L √ 12 2h3/2



a1 a2 a1 + a2

1/2

A list of expressions for the approximate van der Waals interaction energy obtained using Derjaguin’s technique for different geometries is provided in Table 11.2. Recalling the assumptions inherent in Derjaguin’s approximation, primarily the fact that the separation between the interacting bodies must be much smaller than the principal radii of curvature of the bodies, it becomes apparent that Derjaguin’s technique is only applicable for prediction of the van der Waals interaction between quite large objects at very small separations. In fact, the expressions based on Derjaguin’s approximation significantly overestimate the van der Waals interaction energy at separation distances greater than about 20% of the particle radius. 11.2.1.2 Surface Element Integration Starting from the interaction energy per unit area, one can obtain a much better approximation for the van der Waals interaction energy compared to Derjaguin’s technique by using the surface element integration (SEI) (Bhattacharjee and Elimelech, 1997; Bhattacharjee et al., 1998). In this technique, the interaction energy between two bodies is obtained by integrating the interaction energy per unit area over the exact surfaces of the bodies. While computationally somewhat expensive compared to Derjaguin approximation, the procedure can provide the interaction energy more accurately for two bodies of arbitrary geometrical shapes, and even provides the exact Hamaker interaction energy between a body and an infinite flat plate. SEI is thus quite useful in evaluating the van der Waals interactions between a particle and a planar surface, a situation commonly encountered in particle deposition processes. In surface element integration, the interaction energy between two curved surfaces is represented by the integral U (h) =

 

A

Up (H ) n · k dA A |n · k|

(11.22)

11.2

HAMAKER’S APPROACH

401

Here, U is the interaction energy between the two bodies, Up (H )/A is the interaction energy per unit area between the materials of the two bodies, A is the projected area of the body normal to the line of closest distance between the bodies. The term n · k/|n · k| provides a measure of the angle made by the curved surface of the body with the projected normal area A, and assumes a value of +1 or −1 depending on whether the surfaces of the bodies face each other or they face away from each other. When applied for the case of a sphere interacting with an infinite flat surface, SEI provides an analytic expression for the unretarded van der Waals interaction given by 
 a h AH a + + ln U SEI = − (11.23) 6 h h + 2a h + 2a This expression is identical to that for the interaction energy between a sphere of radius a and an infinite planar surface shown in Table 11.1. In other words, SEI provides the exact unretarded van der Waals interaction between a colloid and an infinite planar surface. The results are not exact for two curved surfaces. However, even for such cases, SEI provides remarkably accurate predictions of the interaction energy. Clearly, SEI is much superior to Derjaguin’s approximation when it comes to evaluation of the van der Waals interaction energy between small colloidal particles. The technique can be readily generalized for surfaces of arbitrary geometries (Bhattacharjee et al., 1998; Hoek et al., 2003). 11.2.2 Cohesive Work and Hamaker’s Constant The concept of cohesion work can be utilized to estimate the Hamaker constant, A12 , between bodies 1 and 2. The work of cohesion consists of producing two new interfaces from a given material. It measures the attraction between the molecules of the two portions being produced. The cohesion work due to the dispersion force for material 1 is given by d W11 = 2γ d

(11.24)

where γ is the theoretical surface tension due to dispersion energies. For a given material the attraction potential is d

UA = −

A11 12π r12

(11.25)

where A11 is the Hamaker constant for material 1 that is being separated to produce two parts and r1 is the intermolecular distance. However, the cohesion work is related to the attraction potential by d W11 = −UA

(11.26)

A11 = 24π γ d r12

(11.27)

and we obtain

402

LONDON–VAN DER WAALS FORCES AND THE DLVO THEORY

Equation (11.27) was originally derived by Fowkes (1964) and was used to calculate the Hamaker constant using surface tension data (Hiemenz and Rajagopalan, 1997; Morrison and Ross, 2002). A system for which the Fowkes equation is apt to work best is for nonpolar materials such as alkanes, where the attractive forces are due to the dispersion forces. For alkanes, with γ d ∼ = 25 mJ/m2 and r1 = 0.2 nm, one obtains −20 A11 = 7.5 × 10 J. For the attractive force between materials 1 and 2, the cohesion work is approximated by d W12 = −UA = −

where



1/2 A12 = 2 γ1d γ2d 2 12π r12

r12 = (r1 r2 )1/2

(11.28)

(11.29)

Here, r12 is the intermolecular distance approximated by the root mean square of the individual intermolecular distances. Equation (11.28) leads to d d 1/2 2 A12 = 24π r12 γ1 γ2

(11.30)

A12 = (A11 A22 )1/2

(11.31)

The above equation can be used jointly with Eqs. (11.27) to (11.29) to develop a useful relation for A12 in terms of A11 and A22 :

Equations (11.27) and (11.31) give the means to evaluate the Hamaker constant for two non-polar materials in vacuum using surface tension data. Values for Hamaker constant of several common materials are given in Table 11.3. 11.2.3 Electromagnetic Retardation The expressions for the attractive forces were derived with the assumption that the speed of light is infinite. The energy of the interaction is usually decreased at large distances because the time of electric field propagation from one body to another body and back is such that the fluctuating electric moments become slightly out of phase. The correction due to the finite speed of light is called the retardation correction. This correction is unity for h = 0 and it decreases monotonically with increasing h. For the case of two parallel plates, Gregory (1981) gives the retarded potential energy per unit area as Up A12 =− A 12π h2




1 1 + 5.32(h/λ)



(11.32)

where λ = 10−7 m (London wavelength). Note that the bracketed term of Eq. (11.32) is unity for h = 0 and tends to 0 as h → ∞. For the case of two equal-sized spheres,

11.3

EFFECTS OF INTERVENING MEDIUM

403

TABLE 11.3. Commonly Reported Hamaker Constants, A11 , for Unretarded Interaction for Some Common Materials in Vacuum. Substance

Hamaker Constant A11 J (×1020 )

Substance

Hamaker Constant A11 J (×1020 )

Acetone Aluminum Benzene Carbon tetrachloride Cyclohexane Decane Dodecane Ethanol Ethyl acetate Gold Graphite n-Hexadecane Hexane Magnesia Methyl ethyl ketone Mica

4.10, 4.20 14.0, 15.4, 15.5 5.0 4.78, 5.5 4.82, 5.20 5.45 5.0, 5.84 4.2 4.17 37.6, 45.3, 45.5 47.0 5.1 4.32 10.5 4.53 10.0, 10.8

Octane Oxides (most) Pentane Polybutadiene Polydimethylsiloxane Polyethene oxide Polyisobutylene Polypropylene oxide Polystyrene Polyvinyl acetate Polyvinyl chloride Quartz Rutile (TiO2 ) Silicon Silver Toluene Water

4.5, 5.02 10.0–20.0 3.8, 3.94 8.20 6.27 7.51 10.10 3.8, 3.95 6.4, 9.80 8.91 7.5 7.93 43 25.5 39.8, 40.0 5.40 3.7, 4.35, 4.38

Source: Adapted from Ross and Morrison (1988); Morrison and Ross (2002).

the effect of retardation on the interparticle dispersion potential is given by Schenkel and Kitchener (1960) as
 aA12 1 U =− p ≤ 0.57, h/a ≪ 1 (11.33) 12h 1 + 1.77p U =−

aA12 h




2.17 2.45 0.59 − + 60p 180p 2 420p 3



p ≥ 0.57, h/a ≪ 1

(11.34)

where p = 2π h/λ. Other expressions are given by Schenkel and Kitchener (1960), Gregory (1981), and Hunter (1991), and several of these are summarized in Elimelech et al. (1995). 11.3

EFFECTS OF INTERVENING MEDIUM

Hamaker’s approach for the evaluation of the dispersion force is used when the medium between the two materials is vacuum. A correction to the Hamaker constant is needed when the intervening medium is a dielectric. A common approximation is to estimate the effect of an intervening medium 1 between two bodies of composition 2 and 3 by considering a pseudo-chemical reaction as shown: {2} (1) + (1) [3] → {2} [3] + (1) (1)

(11.35)

404

LONDON–VAN DER WAALS FORCES AND THE DLVO THEORY

where the numbers enclosed within different types of brackets represent different materials. The change in potential energy φ for the case of the above reaction is φ = φ23 + φ11 − φ21 − φ13

(11.36)

Each term on the right side of Eq. (11.36) depends in the same way on the size and distance parameters and differs only in molecular parameters which are fully contained in the Hamaker constant (Hiemenz and Rajagopalan, 1997). One can, therefore, write (11.37) A213 = A23 + A11 − A21 − A13 , where A213 is the overall Hamaker interaction parameter for bodies 2 and 3 being intervened by medium 1. As was pointed out earlier, the interaction between dissimilar bodies is given by the geometrical mean of the homogeneous interactions for the bodies; i.e., A12 = (A11 A22 )1/2 , (11.38) where A11 is the overall Hamaker constant for two bodies of the same material in vacuum. Making use of Eq. (11.38), A213 of Eq. (11.37) becomes A213 = (A22 A33 )1/2 + (A11 A11 )1/2 − (A11 A22 )1/2 − (A11 A33 )1/2

(11.39)

or 1/2 1/2

1/2 1/2 A213 = A33 − A11 A22 − A11 .

For the special case of A22 ≡ A33 , Eq. (11.39) gives 1/2 1/2 2 A212 = A11 − A22

(11.40)

Equation (11.40) defining A212 indicates that for all values of A11 and A22 , the parameter A212 is always positive. This means that the potential energy U is always negative (i.e., attractive) for two similar bodies with an intervening medium. However, the general expression (11.39) shows that A213 can have negative values. Here, a repulsive potential can develop between two dissimilar bodies with an intervening medium. The dispersion force between a particle and an air bubble (air–water–solid) can therefore be repulsive depending on the particle and the intervening medium. Table 11.4 shows values of the Hamaker constant for interaction between common materials when the intervening medium is water. In the literature, it is sometimes customary to lump the Hamaker parameter with the term due to the retardation effects as one variable denoted as Aret (h), with Aret (0) being the Hamaker constant with no retardation effects. Thus, Aret (0) is identical to the term AH used earlier for the effective unretarded Hamaker constant. Table 11.5 gives values for AH = Aret (0) for various materials. Figure 11.3 depicts variations of Aret with distance h for polystyrene parallel plates in pure water and salt water. Aret (h) accounts for the retardation effects, where the speed of light is not taken to be infinity.

11.3

EFFECTS OF INTERVENING MEDIUM

405

TABLE 11.4. Effective Hamaker Constants A212 for Unretarded Interaction Between Common Materials (denoted by 2) Immersed in Water (denoted by 1). Substance

Hamaker Constant A212 J (×1020 )

Alkanes: Pentane Hexane Heptane Octane Decane Dodecane Tetradecane Hexadecane

0.336 0.360 0.386 0.410 0.462 0.50 0.514 0.50

Air Alumina Calcite Copper Crystalline quartz

3.7 4.12, 4.17 2.23 17.5 1.70

Substance

Hamaker Constant A212 J (×1020 )

Fused quartz Fused silica Germanium Gold Magnesia Metals Mica Oxides

0.63,0.833 0.849 16.0, 17.7 27, 33.4, 33.5 1.60, 1.76 3–33.4 2.0, 2.33 1.76–4.17

Polymethyl methacrylate Polystyrene Polyvinyl chloride Quartz Rutile (TiO2 ) Sapphire Silicon Silver

1.05 0.27–1.01 1.30 1.08 26 5.32 11.9, 13.4 26.6, 28.2

Source: Adapted from Ross and Morrison (1988); Morrison and Ross (2002).

TABLE 11.5. Effective Hamaker Constants AH for Unretarded Interaction Between Common Materials in Vacuum (A22 ) and in Water (A212 ). AH (×1020 ), J Calcite Copper Decane Gold Hexadecane Hexadecane Pentane Polyisoprene Polymethyl methacrylate Polystyrene Poly tetrafluoroethylene Polyvinyl chloride Quartz (crystalline) Quartz (fused) Sapphire Silica (fused) Silver Water

Vacuum

Water

10.1 40 4.8 40 5.2 5.4 3.8 6.0 7.1 7.9 3.8 7.8 8.8 6.5 15.6 6.6 50 3.7

2.23 30 0.46 30 0.54 — 0.34 0.74 1.05 1.3 0.33 1.30 1.70 0.83 5.32 0.85 40 —

Source: Adapted from Russel et al. (1989).

(Parsegian and Weiss, 1981)

(Hough and White, 1980)

406

LONDON–VAN DER WAALS FORCES AND THE DLVO THEORY

Figure 11.3. Calculated values for the effective Hamaker constant for interaction between polystyrene blocks in pure and salt water (Parsegian, 1975).

Noting that the Hamaker constant is calculated differently for different combinations of interacting bodies and intervening media, leading to descriptive nomenclature such as A212 , A132 , A12 , etc., it should be borne in mind that the symbol AH used for the effective Hamaker constant in several expressions in this and subsequent chapters should be calculated appropriately for each combination of interacting materials. 11.4

DLVO THEORY OF COLLOIDAL INTERACTIONS

In a colloidal system, whether colloidal particles flocculate or coalesce depends on the net interaction resulting from the combined attractive and repulsive forces to which the colloidal particles are subjected. For a system where electrostatic and van der Waals forces are dominant, the stability of the system, i.e., whether coagulation occurs or not, is dependent on the electrostatic repulsion due to the presence of the electric double layer and the dispersion attractive forces (London–van der Waals). This concept was developed independently by Derjaguin and Landau (1941) in the USSR, and Verwey and Overbeek (1948) in the Netherlands. It is known as the DLVO theory. In order to appreciate the contribution of potential energy due to the electrostatic and London–van der Waals forces, it is best to consider the sum of the interactive potential energies due to these two types of forces for a case of two flat parallel plates. The repulsive electric double layer force between two parallel plates having similar surface potentials was derived previously in Chapter 5 [see, for instance, Eq. (5.113)] and it is given, with usual Debye–Hückel approximation of low potential, by FR /A = 2ǫκ 2 ψs2 exp(−κh)

(11.41)

11.4

DLVO THEORY OF COLLOIDAL INTERACTIONS

407

where FR /A is the repulsive force ( N/m2 ) due to the surface charge and, ψs is the surface potential. The potential energy per unit area due to the repulsive force is defined by the energy required to bring two plates from infinity to a finite separation distance, h. The potential energy is related to the force by dUR /A FR =− dh A

(11.42)

Integration of Eq. (11.42) leads to 

UR /A

dUR = A

∞



h

∞

−

FR dh A

(11.43)

Setting (UR /A)∞ as zero, Eqs. (11.41) and (11.43) give UR =− A



h

2ǫκ 2 ψs2 exp(−κh) dh

(11.44)

∞

The potential for the case of two parallel plates is then given by (in J/m2 ) UR = 2ǫκψs2 exp(−κh) A

(11.45)

The potential (energy) per unit area due to the dispersion force between two similar plates (2) in a medium (1) is given by UA A212 =− A 12π h2

(11.46)

The total free energy per unit area of the interaction between the two flat parallel plates as a function of separation distance h is given by combining Eqs. (11.45) and (11.46): UR UA A212 U = + = 2ǫκψs2 exp(−κh) − (11.47) A A A 12π h2 The above expression for the total energy of interaction U/A determines the stability of the two parallel plates. This stability analysis for a colloidal dispersion is in essence the DLVO theory for colloidal stability. The total energy U/A given by Eq. (11.47) shows that U/A is dependent on the system properties such as ǫ and A212 and on the “manipulated” physical characteristics of the system as manifested by the electrolyte solution properties represented in the inverse Debye length parameter, κ, as well as the surface electric potential, ψs . In a similar fashion, for κa ≫ 1, we can write the total interaction energy for two equal spheres as aA212 U = 2π aǫψs2 exp(−κh) − (11.48) 12h

408

LONDON–VAN DER WAALS FORCES AND THE DLVO THEORY

Figure 11.4. Total interaction energy curves (solid lines) obtained by the summation of an attractive potential curve (dotted line) with different repulsive potential curves (dashed lines).

where (h + 2a) is the distance between the sphere centers, a is the sphere radius, and κ is the inverse Debye length. Here, the van der Waals interaction is assumed to be unretarded. Figure 11.4 shows three different total interaction energy curves for the case of two parallel plates, each obtained by combining the repulsive electrostatic double layer interaction with an attractive van der Waals interaction. The repulsive energy is an exponential function of the interparticle distance h with a range of the order of κ −1 . On the other hand, the van der Waals interaction energy decreases as the inverse power of the interparticle distance. Consequently, the van der Waals attraction energy dominates at very small (near contact) and very large interparticle distances, whereas the electric double layer repulsion dominates at intermediate distances (Probstein, 2003). In Figure 11.4, curve U (1) represents a well-stabilized colloidal system with a repulsive energy maximum. This represents the energy barrier preventing or hindering the approach of two colloidal particles to contact. If this repulsive maximum is large compared to the thermal energy kB T of the colloidal particles, the system should be stable at least when no other forces (like shear forces) exist. Curve U (3) in Figure 11.4, on the other hand, represents a case where the repulsive barrier is absent, implying that the dispersion is unstable. In this scenario, the colloidal particles will coagulate rapidly, as they will be attracted to a deep attractive energy minimum at contact. Curve U (2) represents the transition between stability and coagulation at the primary minimum. An interesting additional feature of the potential energy versus distance plot is the presence of a secondary minimum at relatively large interparticle distance as given by curve U (2). If this minimum is relatively deep (several kB T values), it should give rise to loose flocs. However, this type of a coagulated dispersion can be easily redispersed by agitation.

11.5

SCHULZE–HARDY RULE

409

Figure 11.5. Influence of electrolyte concentration on the total potential energy of interaction of two spherical particles.

It should be noted that at very small distances, repulsion due to the overlapping of electron clouds (Born repulsion) predominates. Consequently, at such small distances there is a deep minimum (primary minimum) in the potential energy curves. This region of the potential energy curves is shown only for curve U (1) of Figure 11.4. Figure 11.5 shows the influence of the Debye length κ −1 on the total energy interaction of two similar spheres. The variation in κ can be thought of in terms of electrolyte concentration. Large values of κ indicates higher electrolyte concentrations. Figure 11.5 clearly shows that for κ = 108 m−1 a stable colloidal system is obtained. Increasing κ to 109 m−1 gives an unstable system where coagulation will occur. The electrolyte causes the diffuse part of the double layer to compress. When the double layer is reduced in thickness, the colloidal particles coagulate because the particles can get close enough to each other for the London–van der Waals forces to dominate. 11.5

SCHULZE–HARDY RULE

One of the earlier generalizations regarding the effect of added electrolyte is a result known as the Schulze–Hardy Rule (1900) (Hiemenz and Rajagopalan, 1997). This rule states that it is the valence of the ions of opposite charge to the colloidal particle (counterions) that have the main effect in the stability of the colloidal particle. The critical flocculation concentration (CFC) value for a particular electrolyte is essentially determined by the valence of the counterion regardless of the nature of the ions having the same charge as the surface (i.e., coions). The CFC is the critical concentration of electrolyte required to flocculate the colloidal particles. In the above rule it is understood that the electrolyte is a non-adsorbing indifferent electrolyte where its addition has no effect on the surface potential ψs .

410

LONDON–VAN DER WAALS FORCES AND THE DLVO THEORY

The essentials of the Schulze–Hardy rule can be derived from the DLVO theory. The primary maximum of curve U (2) is the demarcation point between stability and instability. At this point UR + UA = 0

(11.49)

d (UR + UA ) = 0 dh

(11.50)

and

Making use of the potential energy between two identical spheres valid for higher surface potential and for a (z : z) electrolyte (Overbeek, 1972), one can write U=


 32π aǫ(kB T )2 aA212 2 zeψs tanh exp(−κh) − z2 e 2 4kB T 12h

(11.51)

with κ=



2e2 z2 n∞ ǫkB T

1/2

where the first term of Eq. (11.51) is the double-layer repulsive potential energy, UR , and the second term is the attractive dispersion potential energy, UA . Application of Eqs. (11.49) and (11.50) to (11.51) leads to having the maximum in U occur at a value equal to κ −1 and ncrit


   49.63 kB T 2 zeψs 4 = 6 3 tanh 4kB T z Ib A212

(11.52)

where ncrit is the number concentration of ions and CFC is given in mol/liter (molarity); i.e., 1 ncrit = CFC (11.53) 1000Na and Ib =

e2 4π ǫkB T

(11.54)

For large potential ψs , tanh(zeψs /4kB T ) becomes ≈1 and CFC ∝

1 z6

(11.55)

which explains the Schulze–Hardy rule (Russel et al., 1989). Here, the coagulating effectiveness of the dominating ion increases with its valency in the proportion of 1 : 64 : 729 for uni-, di-, and trivalent ions, respectively. Thus, if the univalent ion requires 729 units of concentration for coagulation, the divalent ion would require 64 units, and the trivalent ion would require only 1 unit.

11.5

SCHULZE–HARDY RULE

411

For small potentials, tanh

zeψs zeψs ≈ 4kB T 4kB T

(11.56)

and ψs4 (11.57) z2 which also partially explains the Schulze–Hardy rule. Here the surface potential becomes important as well. The effect of the surface potential on the critical flocculation concentration (CFC) for z = 1, 2, and 3 are given in Figure 11.6 (Shaw, 1980). The reason that the valency of the counterions is important and not the valency of the coion can be explained by making use of the Boltzmann distribution. For a positive surface potential, ψs is positive and the counterion distribution is given by 
eψs |z− | (11.58) n− = n∞ exp kB T CFC ∝

where z− is the valency of the counterions. For the case of coions: 
|z− |eψs n− = n∞ exp − kB T

(11.59)

For the case of large ψs , Eqs. (11.58) and (11.59) indicate that n− ≫ n+ within the double layer close to the surface. Hence the counterions become the controlling ions.

Figure 11.6. Coagulation concentrations calculated by taking A212 = 10−19 J, for counterion charge numbers 1, 2, and 3. The colloidal system is predicted to be stable above and to the left of each curve and coagulated below and to the right of each curve (Shaw, 1980).

412

LONDON–VAN DER WAALS FORCES AND THE DLVO THEORY

The DLVO theory predicts that ions having the same valency have the same coagulation capability. However, experimented results indicate that the ion effectiveness is as follows (Hiemenz and Rajagopalan, 1997): • For monovalent cations + + + Cs + > Rb+ > NH + 4 > K > Na > Li

• For monovalent anions − F − > Cl − > Br − > NO− 3 >I

11.6 VERIFICATION OF THE DLVO THEORY Many attempts have been made to verify the DLVO theory taking into account both the dispersion and electrostatic forces. The major difficulty was the limitation due to the roughness of the surfaces used. It was Tabor and Winterton (1969) who first recognized that sheared muscovite mica provides a molecularly smooth surface ideal for such measurements. Figure 11.7 shows the comparison between the experimental observations and theoretical predictions of the retarded Hamaker constants of molecularly smooth mica plates. The experiments were conducted in a vacuum. Allowing for experimental errors, the agreement between the theory and experiment can be brought to within 10%. For the case of the force between mica plates in aqueous electrolyte solutions, Figure 11.8 shows the force between two mica plates divided by the radius of curvature R of the plates (Israelachvili, 1985). Here, the agreement between the experimental

Figure 11.7. Retarded Hamaker constant versus separation for mica plates interacting across a vacuum (Chan and Richmond, 1977): , data from Tabor and Winterton (1969);
, data from Israelachvili and Tabor (1973); ——, calculation (Russel et al., 1989).

11.6 VERIFICATION OF THE DLVO THEORY

413

Figure 11.8. Force measured between mica plates (crossed cylinders) in aqueous electrolyte solutions compared with theoretical predictions; R denotes the radius of curvature of the mica plates (Israelachvili, 1985).

results and the theoretical predictions is excellent. The experimental device used in the measurements is described by Israelachvili and Adams (1978). Such an apparatus is shown in Figure 11.9. The force was measured between the macroscopic surfaces, in a configuration of crossed cylinders at nanometer separations. The surfaces consisted of 1 µm thick mica sheet silvered on the back and glued to quartz pieces with a radius of curvature of 10 mm. The separation was measured optically using a spectrometer. The results shown in Figure 11.8 are restricted to situations where there are no hydration effects arising from the presence of hydrated metal ions on the mica surface. The study of Israelachvili and Adams (1978) and the later studies of Pashley (1981a,b) and of Pashley and Israelachvili (1984) indicated that counterion hydration on the mica surface can give rise to an additional short-range repulsive force between mica surfaces in aqueous solutions of K+ and Na+ of given molarity. This hydration force is not affected by an increase in temperature (up to 65◦ C) and it is completely absent in pure water where H3 O+ is the counterion. Figure 11.10 shows a case where force measurements were made in a 4 × 10−5 M KCl electrolyte solution. The agreement with the DLVO theory is excellent and the primary maximum occurs at a separation distance of 2 mm. However, at higher KCl concentrations, say 3 × 10−4 and 10−3 M, no primary maximum was observed and there appears to be an additional repulsive force which cannot be accounted for by the DLVO theory. If one is to account for the solvent structure, then the total interaction potential energy can be given by U = UA + UR + US

(11.60)

where US is the potential due to the solvent structure. This term is necessitated in order to reconcile experimental measurements with the classical DLVO theory. At present,

414

LONDON–VAN DER WAALS FORCES AND THE DLVO THEORY

Figure 11.9. Schematic drawing of apparatus to measure long-range forces between two crossed cylindrical sheets of mica (of thickness 1 µm and radius of curvature 12 mm) immersed in an electrolyte. By use of white light and multiple beam interferometry, the shapes of and separation between the two mica surfaces may be independently measured. The separation between the two mica surfaces can be controlled by use of two micrometer-driven rods and a piezoelectric crystal tube to better than 0.1 nm (Israelachvili and Adams, 1978).

the lack of an adequate theory for the structure of water prevents theoretical evaluation of the potential US . Its magnitude for mica can be estimated using Israelachvili and Adams type measurements. The difference between the measured total potential and that given by the DLVO theory would then be the US term (Hunter, 1991). In general, discrepancy in the region of 0 < h < 5 × 10−9 m occurs when the term US is non-zero. In aqueous solutions, both positive and negative values of US can arise due to the presence of “hydration forces”, where positive interactions pertain to hydrophilic repulsion and negative interactions are believed to be due to hydrophobic attraction. Both types of interaction decay exponentially with a decay length typically on the order of several nanometers (Hunter, 1991; Xu and Yoon, 1989, 1990). For electrokinetic modeling, where the hydration of the ions in the electric double layer occurs, it may become necessary to modify the Boltzmann distribution and to

11.7

LIMITATIONS OF DLVO THEORY

415

Figure 11.10. Forces measured between mica surfaces in KCl solutions at pH 5.7. 10−3 M; 3 × 10−4 M; 4 × 10−5 M. In 4 × 10−5 M no hydration forces were present, hence the surfaces jumped to a primary minimum from h ≃ 2 nm. At the higher salt concentrations, strong hydration repulsion forces prevented the primary minimum adhesion. The inset shows the short-range forces in more detail (Pashley, 1981b).

assume a variable dielectric constant within the electric double layer. The reader can refer to the studies by Gur et al. (1978a,b) and Guzman–Garcia et al. (1990) for details. In the above discussions, we only dealt with aqueous systems. A non-aqueous medium is usually characterized with a low dielectric constant and poor dissociation of the electrolytes. By and large the ionic strengths are much lower than 10−6 M and the inverse Debye length is fairly large. Electrostatic stabilization in non-aqueous media is reviewed by van der Hoeven and Lyklema (1992). 11.7

LIMITATIONS OF DLVO THEORY

Several approximations were made in arriving at the DLVO theory of colloidal interactions. When one systematically revisits these assumptions and approximations, one might tend to wonder as to why the theory provides such a good qualitative and often quantitative description of stability of such a wide array of charged colloidal dispersions. In this section, we will first outline some common approximations made in arriving at the DLVO theory, which might result in deviations of the predictions based on the theory from reality. Following this, we will briefly discuss some of the

416

LONDON–VAN DER WAALS FORCES AND THE DLVO THEORY

common approaches that have been adopted to overcome these limitations of the DLVO theory. 11.7.1 Major Assumptions in DLVO Model of Colloidal Interactions The fundamental assumption in the DLVO theory is that the total interaction energy between two colloidal entities is comprised solely of the electrostatic double layer interaction and the van der Waals interaction. Furthermore, it is assumed that these two interactions can be computed separately and added up as in Eq. (11.47). Both these assumptions are questionable. First, other types of interactions, for instance, the hydration interactions (hydrophobic attraction and hydrophilic repulsion) mentioned in the previous section may be present in colloidal systems. Secondly, when we compute the electric double layer interactions based on the Poisson–Boltzmann equation, the Boltzmann distribution of the ions is obtained solely on the basis of the electrical potential energy. Strictly speaking, the ionic distribution should be based on the total energy or potential of mean force (Russel et al., 1989). This potential of mean force should consist of both the electrical potential as well as the potential due to the dispersion interactions. There are other types of fundamental assumptions inherent in the DLVO theory, which stem from the manner in which we compute the individual components of the potential. In this theory, the solvent is treated as a continuum, and its presence is accounted for through an effective Hamaker constant and an effective dielectric constant. In particular, the granular nature of the solvent is totally ignored in the theory. Consideration of the solvent explicitly should naturally modify the predictions of the interaction energy (Israelachvili, 1985; Hirata, 2003), particularly when the interacting surfaces are close enough such that the intervening distance is of the order of the solvent molecular dimensions (less than 1 nm). In using the Poisson–Boltzmann model for the electrostatic double layer interaction, we also ignore the finite dimensions of the ions, treating them as point charges. Once again, such an assumption will render the electrostatic double layer interactions based on the Poisson–Boltzmann model inaccurate. Particularly, when the bulk ion concentrations are greater than 0.5 M, the ion distributions around charged interfaces are no more governed by the Boltzmann distribution. Instead, at high salt concentrations, one observes ion density fluctuations with multiple peaks (Kjellander et al., 1992; Kjellander and Greberg, 1998; Hirata, 2003). There is another implication of this assumption. For highly charged confinements of nanometer scale dimensions (like a cylindrical capillary pore), neglecting the finite size of the ions can result in unrealistically high counterion concentrations in the confined domains. For instance, if the surface potential of a wall is −125 mV, the concentration of a univalent counterion (e.g., Na+ ) at the wall will be approximately 150 times greater than the bulk concentration. Now, assuming that the bulk concentration of the electrolyte is 0.2 M, we will have a local ion concentration of about 30 M at the wall (Das and Bhattacharjee, 2004). This is unrealistically high, particularly considering the wall to be that of a narrow capillary. Given that a hydrated sodium ion has a diameter of about 0.4–0.5 nm, one simply cannot pack 30 M sodium ions inside a few tens of nanometer diameter capillary.

11.7

LIMITATIONS OF DLVO THEORY

417

Barring the above fundamental type of assumptions, we make several geometrical simplifications when calculating the DLVO potential. For instance, we assume all surfaces to be geometrically smooth and uniformly charged. Naturally occurring surfaces are inherently rough and charges appear on most surfaces due to functional groups that are discrete. Influence of roughness and chemical heterogeneity of the surfaces on the DLVO interactions is a widely studied area, and over the years numerous approaches for studying the influence of these heterogeneities on the interaction energy have been proposed (Czarnecki, 1986; Elimelech and O’Melia, 1990; Herman and Papadopoulos, 1991; Lenhoff, 1994; Suresh and Walz, 1996; Bhattacharjee et al., 1998; Das and Bhattacharjee, 2004). A common consensus from these studies is that the magnitude and range of the interaction energy become considerably suppressed in presence of roughness. When we compute the two components of the DLVO interaction, we often employ different approximations for calculating the van der Waals and electrostatic double layer components. A common error stems for the use of Derjaguin approximation. Notably, Derjaguin approximation introduces a large error in the computation of the van der Waals interaction (Russel et al., 1989; Bhattacharjee and Elimelech, 1997). The error becomes significantly large as the interparticle separation increases. When we incorporate the retardation effects, quite often the particle-particle retarded van der Waals interaction is obtained employing Derjaguin approximation (Schenkel and Kitchener, 1960). Thus, even though our intention is to correct Hamaker’s results by incorporating retardation for large separations, we actually grossly overestimate the interaction energy by using Derjaguin approximation. The error in the DLVO potential becomes quite large when we combine a reasonably accurate expression for the electrostatic double layer interaction with a highly incorrect estimate of the van der Waals attraction. Finally, in calculating the electrostatic interaction, we generally employ constant potential or constant charge density boundary conditions, which are rarely found in real surfaces. A more realistic boundary condition should be the charge regulation type boundary condition, which arises from the interactions between the ions and the surface functional groups (Behrens and Borkovec, 1999). To summarize, when one applies the DLVO theory, one should note that the theory simply states that the attractive and repulsive interaction potentials exert a combined influence on the stability of a colloidal system. There is no specific recipe for obtaining these components of the interactions. In this context, one should be careful about which expressions for the van der Waals and electrostatic interactions are employed in the DLVO potential, and ensure that these are valid for the problem at hand. For instance, employing an electrostatic double layer potential based on the linear superposition approximation will not be accurate when the interparticle separation is small (implying significant double layer overlap). In this case, the repulsive energy barrier height calculated using the linear superposition approximation will be incorrect. Finally, one should note that the DLVO potential is an approximate potential of mean force between two charged particles suspended in an infinitely large medium. Application of this potential to concentrated colloidal systems without considering the many body effects on phenomena such as coagulation or disorder-order transition (Russel et al., 1989) is untenable. There have been several approaches where different

418

LONDON–VAN DER WAALS FORCES AND THE DLVO THEORY

aspects of the many body interactions between colloidal particles in concentrated suspensions are considered within the framework of the DLVO type potential of mean force. A typical modification of the theory in concentrated dispersions involves consideration of a modified Debye length to account for the large particle volume fractions (Chaikin et al., 1982). There are also approaches that explicitly consider many body interactions through pairwise summation of the DLVO interaction between two particles in integral equation theories (Vlachy and Prausnitz, 1992; Bhattacharjee et al., 1999). Such approaches often predict properties of concentrated colloidal dispersions quite accurately. There are several occasions when the DLVO theory has been criticized as inaccurate. However, the inability of the DLVO theory in predicting the behavior of colloidal systems might be an artifact of either the oversimplified nature of the DLVO potential, which fails to account for the complexities inherent in the system studied, or due to inaccurate usage of the DLVO theory. The inaccurate usage of the theory can result from employing expressions of the electrostatic interaction energies that are valid for high electrolyte concentrations (large κa) to predict the behavior of colloidal systems at low electrolyte concentrations (small κa). Similarly, expressions based on the linearized Poisson–Boltzmann equation, if employed to predict coagulation and deposition behavior of highly charged particles (zeψ/kB T > 3), might result in erroneous predictions. Finally, as mentioned earlier, roughness or charge heterogeneity of real interfaces can significantly modify the interactions from those predicted by the DLVO theory assuming smooth geometrical shapes. Consequently, when the DLVO theory fails to predict colloidal phenomena accurately, one should systematically explore the causes of such failure before considering the model to be inaccurate. The DLVO theory was a significant breakthrough in the field of colloidal stability prediction, and has survived six decades of intense scrutiny. When it is applied to a system that respects all the assumptions inherent in the model, it provides a remarkably accurate prediction of colloidal stability. To quote Ninham (1999), “The genius of DLVO lies not in complicated models that add more and more parameters, but in its extraction of the essential physics of the problem of lyophobic colloid stability.” 11.7.2 When DLVO Theory Falls Short Numerous other interactions are invoked when DLVO theory is unable to resolve the behavior of a colloidal system. The most common interactions that are referred to in this context are hydrophobic or hydrophilic interactions (Israelachvili, 1985; van Oss and Giese, 2004). In addition, depletion interactions, structural forces, steric interactions, etc., have been employed to describe colloid stability for many systems (Israelachvili, 1985). At the outset, it should be mentioned that most of the other forces mentioned earlier are ramifications of neglecting the granularity of the solvent and the ions in the DLVO theory. To assess the nature of these other forces in aqueous media, one should first take a closer look at water, the most common solvent studied in colloid science. Water is a complex solvent. It exhibits hydrogen bonding, and has unusual properties

11.7

LIMITATIONS OF DLVO THEORY

419

arising from its strong dipolar nature. A very simple depiction of water’s anomalous behavior is in its large surface tension of 72.8 mJ/m2 . In Section 11.2.2, we noted that the surface tension of apolar liquids can be related to their Hamaker constants. Employing this approach, the surface tension of water can be computed from its Hamaker constant as 21.8 mJ/m2 . The remaining 51 mJ/m2 (i.e., 72.8–21.8 mJ/m2 ) is unaccounted for. This contribution to the surface tension of water is ascribed to the hydrogen bonding effects (van Oss et al., 1988). Nowhere in the calculation of the van der Waals and electrostatic double layer interactions have these hydrogen bonding interactions been taken into account. When two colloidal particles approach separations less than a few nanometers, the water molecules in the intervening space have to be removed to facilitate closer approach between the particles. If the particles have an affinity toward the water molecules, a large amount of energy must be spent to remove the water. In other words, hydrophilic particles will experience a strong hydrophilic repulsion. On the other hand if the particles do not have an affinity for water, it is energetically favorable to remove the intervening water molecules. This is often referred to as hydrophobic attraction between the particles. Typically, hydrophilic or hydrophobic forces are assigned an exponential decay behavior of the form (van Oss, 1993) 

d0 − d G(d) = G(d0 ) exp λ



(11.61)

Here, G is the hydration free energy per unit area between the two interacting surfaces separated by a distance d, d0 is the minimum equilibrium distance corresponding to contact between the surfaces, and λ is the decay length of the hydration interactions. G(d0 ) is the hydration interaction energy at contact, which is usually obtained from the interfacial tension between the interacting surfaces and the solvent. It should be noted that the exponential decay behavior, including the decay length λ, are completely empirical. Depletion forces can arise due to the finite size of ions. Recall that for a particle with constant surface potential approaching another charged surface, the osmotic pressure contribution to the electrostatic double layer force will vanish upon integration over the particle surface. This happens because the ion concentration at the constant potential surface of the particle will be uniform everywhere. When we consider ions of a finite size, this condition of uniform surface concentration cannot be satisfied if the particle approaches too close to another charged surface. In this case, the ions in the intervening space between the charged surfaces will have to be removed to bring the particle close to the surface. This will set up an imbalance in osmotic stress around the particle surface. The net effect of this imbalance is manifested as an attractive force, often referred to as depletion interaction. Depletion forces are generally prominent in polydisperse colloidal suspensions, where the smaller colloidal particles (or polymeric entities) contribute to the osmotic stresses around larger particles (Russel et al., 1989). Imbalance in the concentration of the smaller particles around the larger particles gives rise to the depletion forces in these systems (Weronski and Walz, 2003; Weronski et al., 2003).

420

LONDON–VAN DER WAALS FORCES AND THE DLVO THEORY

Structural forces and steric interactions are other manifestations of granularity of the solvent and microions in a colloidal system. These types of forces also appear due to adsorbed polyelectrolytes or tethered chains on particle surfaces (Wang and Denton, 2004). When dealing with bio-colloids, such as bacteria, steric forces must be accounted for to accommodate the effects of an exo-polymer (or lipo-polysaccharide, LPS) layer on the bacterial surfaces. Quite often, various other interactions (like hydration, depletion, steric, etc.) are simply added to the DLVO interaction potential to yield the extended DLVO (XDLVO) potential (van Oss, 1993). Equation (11.60) is an example of such an approach. While it is extremely useful in many applications involving polymeric and biological systems, such an approach may be questioned from a fundamental point of view, since all the additional interactions are ramifications of the granularity of the solvent and the ions, or other factors neglected in the formulation of the DLVO theory. In this context, there is a gradual, but systematic shift toward development of better models of colloidal interactions based on explicit consideration of the solvent and ion structures. Computer simulations and statistical mechanics based theories are now common for prediction of the colloidal interactions and forces (Levin, 2002; Hirata, 2003).

11.8

NOMENCLATURE

a, a1 , a2 AH Aret (h) Aret (0) A11 A12 A312 c CFC d d0 FR G h kB ncrit n+ n− n∞ r

radius of a spherical body, m effective Hamaker constant, J Hamaker constant accounting for retardation effects, J Hamaker constant not accounting for retardation effects, (=AH ) J Hamaker constant for similar bodies (or material) in vacuum, J Hamaker constant for bodies 1 and 2 in vacuum, J Hamaker constant for materials 3 and 2 with an intervening medium 1, J material constant in Lennard–Jones equation (ǫD σ 6 ) critical flocculation concentration of electrolyte solution required to flocculate a colloidal system, mol/L separation distance between interacting surfaces, m minimum equilibrium cut-off separation between interacting surfaces, m electrostatic repulsive force, N (N/m2 for the case of parallel plates) Gibbs free energy, J gap between two bodies (surface to surface distance), m Boltzmann constant, J/K critical ion number concentration, m−3 coions number concentration, m−3 (assuming ψs > 0) counterions number concentration, m−3 (assuming ψs > 0) ion number concentration in the bulk solution, m−3 distance separating centers of two bodies, m

11.9

PROBLEMS

421

intermolecular distance, m absolute temperature, K Lennard Jones interaction potential between two atoms, J total interaction potential, J (J/m2 for the case of parallel plates) attractive potential energy due to London–van der Waals forces, J (or J/m2 for the case of parallel plates) electrostatic repulsive potential, J (J/m2 for the case of parallel plates) potential due to ionic structure, J (J/m2 for the case of parallel plates) volumes of bodies 1 and 2, respectively, m3 cohesion work for material 1 due to dispersion forces, J absolute value of the valency for a (z : z) electrolyte valency of counterions (assuming ψs > 0) valency of coions (assuming ψs > 0)

r1 T uLJ U UA UR US V1 , V2 d W11 z z− z+

Greek Symbols γ d , γ1d , γ2d ǫ ǫD ζ κ ρ1 , ρ2 σ ψs

11.9

surface tension due to dispersion energies, N/m dielectric constant of a material, C/mV characteristic energy of dipolar interactions in the Lennard–Jones equation zeta potential, V inverse Debye length, m−1 molecular density of bodies 1 and 2, respectively, mol/m3 diameter of an atom in the Lennard–Jones equation, m surface electric potential, V

PROBLEMS

11.1. Using the values of the unretarded Hamaker constants in vacuum from Tables 11.3 and 11.5, determine the effective Hamaker constants, A121 or A132 , in Joules for the following materials: (a) Two polystyrene particles in water. (b) Polystyrene and poly-tetraflouroethylene in hexane. (c) two polytetrafluoroethylene particles in hexane. What is the implication of a negative value of the effective Hamaker constant? 11.2. Staring from the attractive Lennard–Jones potential and using Hamaker’s approach, derive an expression for the van der Waals interaction energy between two spherical particles of same radius a, with their surfaces separated by a distance h. Clearly show the geometrical details of your calculations. 11.3. Plot the variation of the scaled van der Waals interaction energy, UA /AH , where AH is the effective Hamaker constant, with scaled separation distance between two particles of equal radius, ap , calculated using Hamaker’s approach and Derjaguin’s approximation. How do the two results compare for three different particle radii (ap = 10 nm, 100 nm, and 1 µm)?

422

LONDON–VAN DER WAALS FORCES AND THE DLVO THEORY

11.4. Consider the van der Waals and electrostatic double layer interaction energies per unit area between two infinite planar surfaces to be UA = −

AH 12π D 2

and

64n∞ kB T 2 Z exp(−κD) κ respectively, where D is the separation distance between the surfaces and Z = tanh(zeψs /kB T ), with ψs being the surface potential. For the above interactions, the total interaction potential energy and the force can become zero at some separation distance. The conditions for the energy and the force to become zero are given by UR =

UDLV O = UA + UR = 0 and dUDLV O =0 dD Show that this occurs when κD = 2. 11.10

REFERENCES

Behrens, S. H., and Borkovec, M., Exact Poisson–Boltzmann solution for the interaction of dissimilar charge-regulating surfaces, Phys. Rev. E., 60, 7040–7048, (1999). Bhattacharjee, S., and Elimelech, M., Surface element integration: A novel technique for evaluation of DLVO interaction between a particle and a flat plate, J. Colloid Interface Sci., 193, 273–285, (1997). Bhattacharjee, S., Kim, A. S., and Elimelech, M., Concentration polarization of interacting solute particles in cross flow membrane filtration, J. Colloid Interface Sci., 212, 81–99, (1999). Bhattacharjee, S., Ko, C. H., and Elimelech, M., DLVO Interaction between rough surfaces, Langmuir, 14, 3365–3375, (1998). Chaikin, P. M., Pincus, P., Alexander, S., and Hone, D., BCC-FCC, melting, and reentrant transitions in colloidal crystals, J. Colloid Interface Sci., 89, 555–562, (1982). Chan, D., and Richmond, P., van der Waals forces for mica and quartz: Calculations from complete dielectric data, Proc. Roy. Soc. Lond., 353A, 163–176, (1977). Czarnecki, J., The effects of surface inhomogeneities on the interactions in colloidal systems and colloid stability, Adv. Colloid Interface Sci., 24, 283–319, (1986). Das, P. K., and Bhattacharjee, S., Electrostatic double layer interaction between spherical particles inside a rough capillary, J. Colloid Interface Sci., 273, 278–290, (2004). Elimelech, M., and O’Melia, C. R., Effect of particle size on collision efficiency in the deposition of Brownian particles with electrostatic energy barriers, Langmuir, 6, 1153–1163, (1990).

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REFERENCES

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Fowkes, F. M., Attractive forces at interfaces, Ind. Eng. Chem., 56, 40–52, (1964). Gregory, J.,Approximate expressions for retarded van der Waals interaction, J. Colloid Interface Sci., 84, 138–145, (1981). Gur, Y., Ravina, I., and Babchin, A. J., On the electrical double layer theory. 1. A numerical method for solving a generalized Poisson–Boltzmann equation, J. Colloid Interface Sci., 64, 326–332, (1978a). Gur, Y., Ravina, I., and Babchin, A. J., On the electrical double layer theory. 2. The Poisson– Boltzmann equation including hydration forces, J. Colloid Interface Sci., 64, 333–341, (1978b). Guzman–Garcia, A. G., Verbrugge, M. W., and Hill, R. F., Development of a space-charge transport model for ion-exchange membranes, AIChE J., 36, 1061–1074, (1990). Hamaker, H. C., London–van der Waals attraction between spherical particles, Physica, 4, 1058–1072, (1937). Herman, M. C., and Papadopoulos, K. D., A method for modeling the interactions of parallel flat-plate systems with surface-features, J. Colloid Interface Sci., 142, 331–342, (1991). Hiemenz, P. C., and Rajagopalan, R., Principles of Colloid and Surface Chemistry, 3rd ed., Marcel Dekker, New York, (1997). Hirata, F., Theory of molecular liquids, in Molecular Theory of Solvation, Hirata F. (Ed.), Kluwer, Dordrecht, (2003). Hoek, E. M. V., Bhattacharjee, S., and Elimelech, M., Effect of membrane surface roughness on colloid-membrane DLVO interactions, Langmuir, 19, 4836–4847, (2003). Hough, D. B., and White, L. R., The calculation of Hamaker constants from Lifshitz theory with applications to wetting phenomena, Adv. Colloid Interface Sci., 14, 3–41, (1980). Hunter, R. J., Foundations of Colloid Science, vol. 1., Oxford University Press, Oxford, (1991). Israelachvili, J. N., Intermolecular and Surface Forces, Academic Press, London, (1985). Israelechvili, J. N., and Adams, G. E., Measurement of forces between two mica surfaces in aqueous electrolyte solutions in the range 1–100 nm, J. Chem. Soc., Faraday Trans. 1, 74, 975–1001, (1978). Israelachvili, J. N., and Tabor, D., van der Waals forces theory and experiment, Prog. Surface Membrane Sci., 7, 1–55, (1973). Kjellander, R., and Greberg, H., Mechanisms behind concentration profiles illustrated by charge and concentration distributions around ions in double layers, J. Electroanal. Chem., 450, 233–251, (1998). Kjellander, R., Akesson, T., Jonsson, B., and Marcelja, S., Double layer interactions in monovalent and divalent electrolytes – a comparison of the anisotropic hypernetted chain theory and Monte-Carlo simulations, J. Chem. Phys., 97, 1424–1431, (1992). Lenhoff, A. M., Contributions of surface features to the electrostatic properties of rough colloidal particles, Colloids Surf. A, 87, 49–59, (1994). Levin,Y., Electrostatic correlations: From plasma to biology, Rep. Progr. Phys., 65, 1577–1632, (2002). Lifshitz, E. M., The theory of molecular attractive forces between solids, Soviet Physics JETP, 3, 73–83, (1956). Mahanty, J., and Ninham, B. W., Dispersion Forces, Academic Press, London, (1976). Morrison, I. D., and Ross, S., Colloidal Dispersions, Suspensions, Emulsions and Foams, Wiley Interscience, New York, (2002).

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Ninham, B. W., and Parsegian, V. A., van der Waals forces. Special characteristics in lipid–water systems and a general method of calculation based on the Lifshitz theory, Biophys. J., 10, 646–663, (1970). Ninham, B. W., On progress in forces since the DLVO theory, Adv. Colloid Interface Sci., 83, 1–17, (1999). Overbeek, J. Th. G., Colloid and Surface Chemistry. A Self-Study Course. Part 2, Lyophobic Colloids, Cambridge, Mass., MIT Center For Advanced Eng. Study, (1972). Parsegian, V. A., Long range van der Waals forces, in Physical Chemistry: Enriching Topics in Colloid and Surface Science, Van Olphen, H. and Mysels, K. J. (Eds.) (1975). Parsegian, V. A., and Weiss, G. H., Spectroscopic parameters for computation of van der Waals forces, J. Colloid Interface Sci., 81, 285–289, (1981). Pashley, R. M., Hydration forces between mica surfaces in aqueous electrolyte solutions, J. Colloid Interface Sci., 80, 153–162, (1981a). Pashley, R. M., DLVO and hydration forces between mica surfces in Li+ , Na+ and Cs+ electrolyte solutions, J. Colloid Interface Sci., 83, 531–546, (1981b). Pashley, R. M., and Israelachvili, J. N., DLVO and hydration forces between mica surfaces in Mg2+ , Ca2+ , Sr2+ and Ba2+ chloride solutions, J. Colloid Interfce Sci., 97, 446–455, (1984). Probstein, R. F., Physicochemical Hydrodynamics, An Introduction, 2nd ed., Wiley Interscience, New York, (2003). Ross, S., and Morrison, I. D., Colloidal Systems and Interfaces, John Wiley, New York, (1988). Russel, W. B., Saville, D. A., and Schowalter, W. R., Colloidal Dispersions, Cambridge University Press, Cambridge, (1989). Schenkel, J. M., and Kitchener, J. A., A test of the Derjaguin-Landau-Verwey-Overbeek theory with a colloidal suspension, Trans. Faraday Soc., 56, 161–173, (1960). Shaw, D. J., Introduction to Colloid and Surface Chemistry, 3rd ed., Butterworths, London, (1980). Suresh, L., and Walz, J., Effect of surface roughness on the interaction energy between a colloidal sphere and a flat plate, J. Colloid Interface Sci., 183, 199–213, (1996). Tabor, D., and Winterton, R. M., Direct measurements of normal and retarded van der Waals forces, Proc. Roy. Soc. Lond., 312A, 435–450, (1969). van der Hoeven, P. H. C., and Lyklema, J., Electrostatic stabilization in non-aqueous media, Adv. Colloid Interface Sci., 42, 205–277, (1992). van Oss, C. J., Chaudhuri, M. K., and Good, R. J., Interfacial Lifshitz–van der Waals and polar interactions in macroscopic systems, Chem. Rev., 88, 927–941, (1988). van Oss, C. J., Acid–base interfacial interactions in aqueous media, Colloids Surf. A, 78, 1–49, (1993). van Oss, C. J., and Giese, R. F., Role of the properties and structure of liquid water in colloidal and interfacial systems, J. Dispersion Sci. Technol., 25, 631–655, (2004). Vlachy, V., and Prausnitz, J. M., Donnan equilibrium: Hypernetted-chain study of onecomponent and multicomponent models for aqueous polyelectrolyte solutions, J. Phys. Chem., 96, 6465–6469, (1992). Wang, H., and Denton, A. R., Effective electrostatic interactions in suspensions of polyelectrolyte brush coated colloids, Phys. Rev. E, 70, Art. No. 041404, (2004). Weronski, P., and Walz, J. Y., An approximate method for calculating depletion and structural interactions between colloidal particles, J. Colloid Interface Sci., 263, 327–332, (2003).

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Weronski, P., Walz, J. Y., and Elimelech, M., Effect of depletion interactions on transport of colloidal particles in porous media, J. Colloid Interface Sci., 262, 372–383, (2003). White, L. R., On the Deryaguin approximation for the interaction of macrobodies, J. Colloid Interface Sci., 95, 286–288, (1983). Xu, Z., andYoon, R. H., The role of hydrophobic interactions in coagulation, J. Colloid Interface Sci., 132, 532–541, (1989). Xu, Z., and Yoon, R. H., A study of hydrophobic coagulation, J. Colloid Interface Sci., 134, 427–434, (1990).

CHAPTER 12

COAGULATION OF PARTICLES

12.1

INTRODUCTION

In the previous chapter, we dealt with van der Waals forces and discussed the DLVO theory, which represents the summation of the van der Waals and the electric double layer interactions between colloidal particles. In this context, we were able to discern whether two particles, when brought together, would coagulate. Our arguments were based solely on the variations of the DLVO interaction potential with separation distance between the colloidal particles. Consequently, we were only able to predict that coagulation might occur if the attachment of two particles is energetically favorable. In addressing coagulation in terms of the energetically favorable state, we ignored the dynamic processes involved in bringing the two particles close together. When suspended colloidal particles in a dispersion are not stable, the rate at which they coagulate depends on the frequency with which they encounter, or collide with, each other. This frequency of collision is a function of the velocities of the fluid and the particles, Brownian motion, and colloidal forces, e.g., electrostatic and van der Waals forces. When two particles of comparable size collide with each other, the collision may lead to the formation of a doublet. This process is termed coagulation. Here, it is usually necessary to know the velocity field of the interacting particles and the prevailing interaction forces. On the other hand, when a particle collides with a much larger particle or a surface, usually called a collector, such a process is often termed

Electrokinetic and Colloid Transport Phenomena, by Jacob H. Masliyah and Subir Bhattacharjee Copyright © 2006 John Wiley & Sons, Inc.

427

428

COAGULATION OF PARTICLES

deposition. The study of deposition usually involves a knowledge of the prevailing forces and flow field around the collector. The physics of particle-particle interaction in air as well as in a liquid are similar. Dissimilarities between interactions in aerosols (gas) and hydrosols (liquid) arise from the magnitudes of the prevailing forces and the ratio of the diameter of an aerosol particle to the mean-free path of the gas molecules. For example, charged aerosols are strongly influenced by electrostatic forces at comparatively large distances from a surface. One the other hand, electrostatic forces in an aqueous medium are restricted by the presence of the electrical double layers (Spielman, 1977). For submicron particles, the primary collision mechanism is due to the Brownian motion. When coagulation is caused solely by the Brownian motion, it is referred to as perikinetic coagulation. On the other hand, when coagulation is caused solely by deterministic forces, e.g., hydrodynamic forces, it is called orthokinetic coagulation. For two colloidal particles that are similarly charged, the interparticle interaction potential typically includes an attractive part due to the London–van der Waals forces and a repulsive part due to the electrostatic forces. When the electrostatic repulsive forces are absent, particles undergo coagulation and the coagulation process is termed rapid coagulation. When the electrostatic repulsive forces or any other repulsive forces, e.g., steric forces are present, the process is termed slow coagulation. The rate of coagulation is a measure of the colloidal system stability.

12.2

DYNAMICS OF COAGULATION

The dynamics of coagulation address how rapidly or slowly a suspension of colloidal particles will coagulate. The rate of coagulation has immense practical implications. Addition of alum causes suspended colloidal matter in water to coagulate and settle during water treatment in clarifiers. The rate of coagulation plays an important role in the design of clarifiers in water purification plants. A colloidal dispersion is considered stable if the coagulation rate is extremely slow. On the other hand, a dispersion is considered unstable if the particles in it coagulate rapidly. In a colloidal suspension, the suspended particles are in a state of continuous random motion. It is conceivable that such a random motion of particles will result in collision between the particles. The probability of a collision can be increased by stirring the suspension vigorously. If there is an electrostatic repulsion between the particles, then the repulsive interaction will hinder the collision between the particles. On the other hand, if the electrostatic repulsion does not exist, the particles become “sticky” due to the van der Waals attraction. In such a scenario, all the collisions between two particles will be “successful”, resulting in the formation of doublets, triplets, quadruplets, and successively larger aggregates. Brownian motion, hydrodynamic interactions, and other interparticle interactions determine the rate of collision between particles. Smoluchowski (1917) was the first to study the rate of coagulation of particles due solely to Brownian motion with the diffusion coefficient of the particles taken as constant. His model was essentially that of diffusion of non-interacting spheres relative to a reference sphere. The

12.3

BROWNIAN MOTION

429

diffusion dynamics were described by the radial component of the diffusion equation in spherical coordinates with a uniform diffusivity (Feke and Schowalter, 1983). Smoluchowski evaluated the rate of diffusion of the particles to a reference sphere and derived an expression for the collision frequency. Smoluchowski also evaluated the collision frequency of spherical particles in a shear flow in the absence of the Brownian motion. Here, it was assumed that the particles follow undisturbed streamlines and that no interparticle forces are present. Comparison of the collision frequency of the two asymptotic cases clearly demonstrated the enhancement of coagulation due to shear flow. Fuchs (1934) generalized Smoluchowski’s analysis to include an arbitrary interaction potential, assuming again that the diffusion coefficient of the particles is constant. Derjaguin and Landau (1941) and Verwey and Overbeek (1948) incorporated electrostatic and van der Waals forces into the arbitrary interaction potential derived by Fuchs to show the effect of these forces on the collision frequency between colloidal particles. Further improvements of the Fuchs derivation were made by Spielman (1970) and Honig et al. (1971), who included the variation of the particle diffusion coefficient due to the presence of another particle. Subsequently, van de Ven and Mason (1976), Zeichner and Schowalter (1977), and van de Ven (1989) evaluated collision frequency, in the absence of the Brownian motion, by incorporating hydrodynamic interactions and interaction potentials into the flow equations. In the following sections, we shall explore, in some detail, the historical developments together with the more recent studies. 12.3

BROWNIAN MOTION

The coefficient of diffusion, D, is an important transport property of a colloidal particle and it represents the role of Brownian motion on the movement of the particle. One normally associates the diffusion coefficient with Fick’s law, j∗∗ D = −D∇n

(12.1)

where j∗∗ D is the particle diffusive flux, and ∇n is the gradient of the number concentration of particles. In other words, one typically associates diffusion with the transport of a particle from a region of high concentration to a region of low concentration due to Brownian motion. This, however, does not imply that diffusion and Brownian motion cease to exist when there is no concentration gradient. Fick’s law simply states that a directional flux of particles is developed due to diffusion when there is a concentration gradient. When the concentration gradient vanishes, the flux also vanishes. However, Brownian motion still exists in a suspension of uniform particle concentration. The random Brownian motion still causes a particle to “wander” around in the suspension volume, but in a macroscopic sense, this does not cause a change in the concentration of the particles. When considering diffusion of a colloidal particle, it should be noted that the random Brownian motion is imparted to the colloidal particle by the thermal motion of the solvent molecules.

430

COAGULATION OF PARTICLES

Figure 12.1. Diffusion in a one-dimensional plane.

The coefficient of diffusion D can be expressed in terms of the particle size. Consider one-dimensional diffusion of a swarm of colloidal particles. Assume that such a swarm is released over a narrow slit at x = 0 and t = 0 such that the number of released particles is Ni per unit area. The number of released colloidal particles per unit area is Ni . At time t = 0, the concentration of the particles for x = 0 is zero. With increasing time, the colloidal particles will diffuse due to the Brownian motion within the medium. Figure 12.1 shows the spread of the particles at time t in one dimension. The governing equation for the spread of the particles is given by the convection-diffusion equation, while dropping out the convection term, ∂n ∂ 2n =D 2 ∂t ∂x

(12.2)

where n(x, t) is the number of particles per unit volume at distance x and at time t. The boundary conditions are n(∞, t) = 0

for all t

and n(−∞, t) = 0

for all t

Initially all the particles are located at x = 0 and n(x, 0) = 0 for |x| > 0. The solution of Eq. (12.2) is given by the Gaussian distribution, viz.,
 x2 Ni exp − n(x, t) = 2(πDt)1/2 4Dt

(12.3)

12.3

431

BROWNIAN MOTION

where Ni is the number of particles released at x = 0 per unit cross-sectional area. It is given by  ∞ Ni = n(x, t)dx −∞

The mean square displacement at time t of the particles from x = 0 is given by x2 =

1 Ni



∞

x 2 n(x, t)dx

(12.4)

−∞

If one makes use of the expression for n(x, t) given by Eq. (12.3), the mean square displacement becomes x 2 = 2Dt

(12.5)

Equation (12.5) states that the mean square displacement of the diffusing particles is proportional to the lapsed time t. For a particle of mass m with velocity v, one can use Newton’s laws of motion to write a force balance as dv = −f v + F (t) (12.6) m dt an equation known as the Langevin equation. Equation (12.6) balances the inertial force arising from the acceleration (dv/dt) against the dissipative force represented by −f v (where f is the friction coefficient) and a stochastic thermal noise F (t) of the fluid around the particle. At infinite dilution and in the absence of charge, the friction coefficient for spherical particles of radius a in a fluid of viscosity µ is given by the Stokes equation f = 6π µa

(12.7)

where, as one can readily see, the units of f are force per unit velocity, i.e., Ns/m. The second term on the right hand side of Eq. (12.6) represents the fluctuating force resulting from the thermal motion of the molecules of the fluid. The random trajectories of the particles are the result of the random fluctuation in the forces that the colliding fluid molecules collectively exert on the particles. The fluctuating force F (t) is assumed to be independent of the particle velocity v and its mean value over a long time t is zero. Equation (12.6) can be written as d(vx) + βvx = v 2 + xA(t) dt

(12.8)

where β = f/m, A(t) = F (t)/m, and v = dx/dt. Integrating Eq. (12.8) between t = (0, t) and recognizing that at t = 0, vx = 0, one obtains vx = exp(−βt)



0

t

2

v exp(βτ ) dτ + exp(−βt)



t

A(τ )x exp(βτ ) dτ 0

(12.9)

432

COAGULATION OF PARTICLES

where τ is a variable of integration. Making use of averaging arguments, Friedlander (2000) gave at large times x2 v2 t = 2 β

(12.10)

We can now make use of a physical argument made by Einstein that relates the Brownian motion of a colloidal particle to the molecular motion of the medium. As the particles share the molecular-thermal motion of the fluid, the principle of equipartition of energy is assumed to apply to the translational energy of the colloidal particles and leads to kB T mv 2 = (12.11) 2 2 Combining Eq. (12.10) with (12.11) and making use of the definition of β allows one to relate the mean-square deviation in position to the viscous dissipation x2 = 2

kB T t f

(12.12)

A comparison of Eq. (12.5) with (12.12) reveals that D=

kB T f

(12.13)

the well-known Stokes–Einstein equation for the coefficient of diffusion of an inert spherical particle. A discussion on the various forms of the diffusion coefficient as influenced by the proximity of other particles, the particle surface potential, the Debye length, and the size of the ions is given in van de Ven (1989). At infinite dilution, the diffusion coefficient of an inert particle in a fluid is given by D = D∞ =

kB T 6π µa

(12.14)

where kB is Boltzmann constant and T is the absolute temperature. In the above analysis, the diffusion coefficient D was assumed to be constant and not affected by the presence of surrounding particles.

12.4

COLLISION FREQUENCY

In this section, we shall derive the kinetic equation for coagulation. Here particle collisions lead to coagulation and hence a reduction in the total number of particles and an increase in the particle average size. We shall assume that every collision leads to coagulation. Consider a colloidal dispersion containing an infinite number of species (i = 1, 2, 3, . . . , ∞) characterized solely by its volume vi . Let Jij be the collision

12.4

COLLISION FREQUENCY

433

frequency per unit volume occurring between two classes of spherical particles characterized by volume vi and vj , respectively. When particle i collides with particle j a third particle is formed having a volume of (vi + vj ). The collision frequency per unit volume Jij can be given as Jij = β(vi , vj )ni nj

(12.15)

where ni is the number concentration of species i. Here β(vi , vj ) is the collision frequency function. This function is dependent on the system properties and on the prevailing interparticle forces. The rate of formation per unit volume of particles of size k by collision of particles of size i and j is given by 1  Jij (12.16) 2 i+j =k The notation i + j = k means that the summation is taken over those collisions for which a volume vk can be produced; i.e., (12.17)

vi + vj = vk

The factor 1/2 in Eq. (12.16) is present because each collision is counted twice in the summation. Equation (12.16) is due to the generation of the k th species. In turn, particle k collides with other particles leading to particle volumes larger than vk . The disappearance term is given by ∞  Jik (12.18) i=1

The net balance equation for the k species is given by th

dnk 1 = dt 2

i=k−1 

i=1;i+j =k

Jij −

∞ 

Jik

i=1

for k = 1, 2, 3, . . . , ∞

(12.19)

With the use of Eq. (12.15), the above equation becomes dnk 1 = dt 2

i=k−1 

i=1;i+j =k

βij ni nj − nk

∞  i=1

βik ni

for k = 1, 2, 3, . . . , ∞

(12.20)

where βij = β(vi , vj ). Equation (12.20) is the dynamic equation for the evolution of the population of particle k with time. This equation, called the population balance equation, is a fairly general equation that accounts for the generation and loss of a population. The solution of Eq. (12.20) depends on the collision frequency function βij , which is determined solely by the mechanism of the particle collisions which is dependent on the prevailing interparticle forces.

434

12.5

COAGULATION OF PARTICLES

BROWNIAN COAGULATION

12.5.1 The Smoluchowski Solution without a Field Force First, we shall deal with the diffusion of colloidal particles due to their Brownian motion. To begin with, the particles are assumed to be non-interacting and the only prevailing forces are due to the Brownian motion. In other words, forces due to electrostatic and dispersion are not accounted for. This analysis was put forward by M. von Smoluchowski in the early 1900s. Consider a diffusional flux of spherical particles of volume vj diffusing towards a particle of volume vi . Particle vi is the reference or test particle fixed at the coordinate system. In the absence of a convective flux, the diffusion equation is given by
 ∂nj 1 ∂ 2 ∂nj =D 2 r (12.21) ∂t r ∂r ∂r where nj is the number concentration of particle vj and r is the radial coordinate of the spherical coordinate system. Angular symmetry is assumed. The initial condition is given as At t = 0

nj = nj ∞

The boundary conditions are at r = ai + aj

for r > ai + aj

nj = 0

for t > 0

and at r → ∞ nj = nj ∞

for t > 0

Here ai is the radius of the particles having volume vi and nj ∞ represents the uniform number concentration of the j th particle. One can solve Eq. (12.21) by making use of the transformation (Friedlander, 2000) 

nj ∞ − nj r (12.22) w= nj ∞ ai + aj and y=

r − (ai + aj ) ai + aj

(12.23)

The transformed form of Eq. (12.21) is

∂w ∂ 2w = D′ 2 ∂t ∂y where D ′ = D/(ai + aj )2 . The solution of Eq. (12.24) is given by 
y w = 1 − erf 2(D ′ t)1/2

(12.24)

(12.25)

12.5

BROWNIAN COAGULATION

435

where erf is the function which has the property of erf(0) = 0. At t → ∞, Eq. (12.25) gives   a i + aj nj = nj ∞ 1 − (12.26) r This is the steady-state solution which can be obtained by setting ∂nj /∂t = 0 in Eq. (12.21). It is interesting to note that steady state is reached in relatively short times (Probstein, 2003). For particles of radius a, one may define the characteristic time for Brownian diffusion as a2 a2 6π µa 3 = = D [kB T /(6π µa)] kB T

(12.27)

For a colloidal particle with a = 10−7 m in water at room temperature, the characteristic time is about 5 × 10−3 s indicating that steady state is reached relatively quickly. The total rate (particles per unit time) at which particles j arrive at the surface r = ai + aj is given by 

D 4π r

2 ∂nj

∂r



r=ai +aj

= 4π(ai + aj )Dnj ∞



ai + aj 1+ (πDt)1/2



(12.28)

The steady-state rate at which particles j arrive at the collision radius of the test particle i is given by 4π(ai + aj )Dnj ∞ The collision rate frequency per unit volume between particles i and j is then given by Jij = 4π(ai + aj )Dni∞ nj ∞

(12.29)

Making use of Eq. (12.15), βij for the case of the Brownian coagulation is given by βij = 4π(ai + aj )D

(12.30)

It is necessary now to quantify the diffusion coefficient D. When the test particle i is also in Brownian motion, the diffusion coefficient D should be relative to the motion of the two particles given by (xi − xj ) where xi and xj are the displacement of the particles in x-direction. From Eq. (12.5), one can write (xi − xj )2 = 2Dt

(12.31)

Equation (12.31) can be expanded to give xi2 − 2xi xj + xj2 = 2Dt

(12.32)

436

COAGULATION OF PARTICLES

As the motions of the two particles are independent of each other, xi xj = 0. Equations (12.15) and (12.32) give Di + Dj = D (12.33) Making use of Eqs. (12.29) and (12.33), we can write for the collision rate per unit volume Jij = 4π(ai + aj )(Di + Dj ) ni∞ nj ∞

(12.34)

The Stokes–Einstein relation, Eq. (12.14), when combined with Eq. (12.34) gives 


2kB T 1 1 Jij = (ai + aj ) + ni∞ nj ∞ 3µ ai aj

(12.35)


1 1 2kB T (ai + aj ) + βij = 3µ ai aj

(12.36)

and

For convenience, the ∞ subscript will be dropped and ni and nj become the prevailing concentrations. Substituting for βij from Eq. (12.36) in the population balance, Eq. (12.20) yields the kinetic equation describing the change in the population of the k th particle. Equation (12.35) represents the collision rate caused solely by the Brownian motion. In order to obtain an analytical solution to Eq. (12.20), a simplifying assumption is necessary. Smoluchowski assumed that ai = aj , which leads to (ai + aj )




1 1 + ai aj



=4

(12.37)

For an initially monodispersed colloidal system, the assumption of ai = aj is valid. However, at long times it is not true. To a large extent, its justification rests on experimental evidence. Making use of Eq. (12.37), the collision frequency function of Eq. (12.36) becomes 8kB T =K (12.38) β(vi = vj ) = 3µ where K is a constant independent of particle size. The population balance equation, (12.20), becomes dnk K = dt 2

i=k−1 

i=1;i+j =k

ni nj − Knk

∞ 

ni

i=1

k = 1, 2, . . . , ∞

(12.39)

Let the total number of particles per unit volume in a closed system at time t be Ntot (t); then ∞  Ntot (t) = ni i=1

12.5

BROWNIAN COAGULATION

437

Summing Eq. (12.39) over all particles k = 1, 2, . . . , ∞, one obtains ∞ K dNtot = dt 2 k=1

i=k−1 

i=1;i+j =k

2 ni nj − KNtot

2 The first term on the right side of the above equation is given by (K/2)Ntot , leading to

dNtot K 2 = − Ntot dt 2

(12.40)

The above equation for the Brownian coagulation is analogous to second-order reaction kinetics. The condition is given by Ntot = N0 at t = 0. Integration of Eq. (12.40) leads to N0 Ntot (t) = (12.41) 1 + KN0 t/2

The above equation shows the decay in the total number of particles with time. Making use of Eq. (12.39), it can be shown (e.g., Friedlander, 2000) that nk (t) = N0 where tBr =

(t/tBr )k−1 (1 + t/tBr )k+1

3µ 2 = 4N0 kB T KN0

(12.42)

(12.43)

The characteristic time tBr is known as the coagulation time and it is the time for the concentration to halve itself (Zeichner and Schowalter, 1979; Probstein, 2003). Figure 12.2 shows the variation of nk /N0 for k = 1, 2, . . . , 4 with dimensionless time, t/tBr . It is interesting to note that (Ntot /N0 ) and (n1 /N0 ) decay monotonically; however, (nk /N0 ) with k ≥ 2 exhibits a maximum before its final decay. One can rewrite the expression for the characteristic time tBr as tBr =

π µa 3 α p kB T

(12.44)

where αp is the colloidal particles volume fraction. Table 12.1 shows the time scale for the formation of a doublet under Brownian motion. It is clear from Table 12.1 that Brownian coagulation is not important for particles much larger than 10 µm where the characteristic time becomes fairly large. 12.5.2 Effect of a Field Force Fuchs (1934) was the first to introduce an interparticle field force to the Smoluchowski Brownian coagulation analysis. Once again, we consider a test particle of radius ai to which particles aj are diffusing. Here particles i and j exert a force on each other.

438

COAGULATION OF PARTICLES

Figure 12.2. The variations in Ntot , n1 , n2 , . . . with time for an initially monodisperse colloidal system. The total number concentration, Ntot , and the concentration of the species n1 both decrease monotonically with increasing time. The concentrations of n2 (t), n3 (t), n4 (t), etc. pass through a maximum (Smoluchowski, 1917).

Let this force be F (r) and it is a function of position. The steady-state collision rate per unit area between particles j and the surface of the test particle i is given by   dnj F (r) + nj − −D dr f The diffusion coefficient D is relative to the motion of the two particles and for an infinite medium it is given by Eqs. (12.14) and (12.33). The total collision rate (particle per unit time) between particles j and the test particle i is given by 

 dnj F (r) − −D + nj 4π r 2 = constant dr f TABLE 12.1. Characteristic Time, tBr for Brownian Coagulation of Spherical Particles in Water at 293 K for αp = 0.1. a (m)

tBr (s)

10 10−7 10−6 10−5

7.77 × 10−6 7.77 × 10−3 7.77 7.77 × 103

−8

12.5

439

BROWNIAN COAGULATION

Defining the force F (r) by its potential U (r) leads to   dnj 1 dU + nj 4π r 2 = constant D dr f dr Making use of Eq. (12.13) one obtains for the total collision rate between particles j and the test particle i,   nj dU dnj 2 + = constant (12.45) 4π r D dr kB T dr Differentiation of Eq. (12.45) with respect to r gives  
nj dU dnj d 2 + r D =0 dr dr kB T dr

(12.46)

where nj = 0 at r = ai + aj and nj = nj ∞ at r =→ ∞ Solution of Eq. (12.46) subject to the above boundary conditions gives 


 r    ∞   U (r) U (r) dr U (r) dr nj = nj ∞ exp − exp exp kB T kB T r 2 kB T r 2 ai +aj a1 +a2 (12.47) The total rate at which particles j arrive at surface r = ai + aj is given by   dni D 4π r 2 dr r=ai +aj Making use of Leibnitz’s rule for the differentiation of integrals, the collision rate per unit volume between particles i and j is then given by   4π D(a + a ) i j   ni∞ nj ∞

Jij =  (12.48) ∞ 1 (ai + aj ) ai +aj r 2 exp UkB(r) dr T

The term in the square brackets in βij . The denominator of the term within the square brackets is the stability ratio W . Making use of Eq. (12.33) and the Stokes–Einstein equation for Di , setting ni∞ = nj ∞ = n, ai = aj = a for the case of a monodisperse system and putting J = Jij , one obtains J =

8kB T · ∞ 3µ 2a 2a

1 1 r2

exp

U (r) kB T



(12.49) dr

For coagulation solely due to Brownian motion, i.e., in the absence of any external force, Eq. (12.35) can be written for ai = aj = a and ni∞ = nj ∞ = n as J0,Br =

8kB T 2 n 3µ

(12.50)

440

COAGULATION OF PARTICLES

where J0,Br denotes the Smoluchowski collision rate per unit volume due solely to Brownian motion, Combining Eqs. (12.49) and (12.50) leads to W =

J0,Br J

(12.51)

where W is the stability ratio. For the case of constant diffusion coefficient, it is given as
  ∞ 1 U (r) W = 2a exp dr (12.52) 2 kB T 2a r When U (r) is a positive function, it signifies repulsion between the particles and it leads to J < J0,Br where W > 1. A large value for W means a stable system. In general, the potential U (r) can have the form of the London–van der Waals (attractive) potential or an electrostatic (repulsive) potential. W is equal to unity for U = 0. Figure 12.3 shows the effect of attractive London–van her Waals forces on the rapid Brownian coagulation J , normalized with the Smoluchowski Brownian coagulation J0,Br . When AH /kB T = 0, the stability ratio is unity and it decreases with AH /kB T indicating rapid coagulation. However, the results as presented in Figure 12.3, which are based on the analysis presented here, are not in agreement with the experimental data. The major simplifying assumption lies in the fact that the diffusion coefficient is taken to be constant and independent of the gap width between the two particles. Recall the discussion in Section 6.3 (e.g., Eq. 6.138) regarding the modification of the particle

Figure 12.3. Variation of stability ratio W with AH /kB T . The van der Waals interaction potential is given by UA = −AH a/[12(r − 2a)] = −AH /(12h). Here, AH is the Hamaker constant. The diffusion coefficient is a constant.

12.5

BROWNIAN COAGULATION

441

hydrodynamic velocities and diffusion coefficients owing to hydrodynamic interactions. For finite size colloidal particles, incorporation of the hydrodynamic interaction effects becomes important. Incorporation of these hydrodynamic interactions in the model for Brownian coagulation is briefly elaborated next. Independently, Derjaguin and Muller (1967), Spielman (1970), and Honig et al. (1971) modified the analysis of the Brownian motion coagulation with interparticle potential to account for the decrease in the diffusion coefficient which occurs when two particles are very close. = The diffusion coefficient tensor D for the case of two spheres is given by Brenner (1961) and Van de Ven (1989) as   = d (h) 0 D = (Di + Dj )∞  (12.53) 0 d⊥ (h) where d (h) and d⊥ (h) are correction factors for the diffusion coefficient of sphere i relative to sphere j . Symbols  and ⊥ signify the relative diffusion parallel and normal to the two surfaces, respectively. The correction factors, d (h) and d⊥ (h), are due to hydrodynamic interactions between the two spheres (van de Ven, 1989). Here d (h) signifies the effect of hydrodynamic interactions on the diffusion coefficient due to the motion of the spheres parallel to each other. This component acts perpendicular to the line joining the centers of the two spheres. The term d⊥ (h) signifies the effect of hydrodynamic interactions on the diffusion coefficient due to the motion of the spheres along the line joining their centers. Figure 12.4 schematically depicts the directions of the two components of the hydrodynamic correction to the diffusion coefficient. The terms Di∞ and Dj ∞ are the particle diffusion coefficients at infinite dilution and they are given by Eq. (12.14). For the case of spheres of equal radii, we set d⊥ (h) = 1/G(h). Here h is the dimensionless gap width between the sphere, where r =2+h a

with ai = aj = a

and r is the dimensional distance between the sphere centers. G(h) is a complex function of h. A simplified form for G(h) is given by Honig et al. (1971) for the case of equal-sized spheres: 6h2 + 13h + 2 G(h) = (12.54) 6h2 + 4h The above expression indicates that G(h) → 1

as h → ∞

and G(h) → 1/(2h) as h → 0

For h → 0, the diffusion coefficient D becomes much smaller than D∞ indicating a retardation in the Brownian coagulation process.

442

COAGULATION OF PARTICLES

Figure 12.4. Schematic depiction of the directions of the hydrodynamic correction components of the diffusion tensor (a): d (h) and (b): d⊥ (h). The arrows indicate the direction of motion in each case. The dashed line represents the line joining the sphere centers.

Following an approach similar to that adopted previously, the coagulation rate per unit volume, for equal-sized spherical particles having a variable diffusion coefficient, becomes

 −1   ∞ G(h) exp U (h) kB T 8kB T  J = 2 dh n2 (12.55) 2 3µ + h) (2 0 with

W =

J0,Br J

(12.56)

where W is modified to W =2



∞

0

G(h) exp

U (h) kB T

(2 + h)2



dh

(12.57)

The characteristic time becomes tBr =

π µa 3 W 3µW = α p kB T 4N0 kB T

(12.58)

where αp is the volume fraction of the particles and N0 is their initial number concentration.

12.5

BROWNIAN COAGULATION

443

If one is to assume that U (h) = 0, i.e., a complete absence of an external potential and that the diffusion coefficient is affected by the gap width, then W → ∞ indicating that Brownian motion alone cannot lead to coagulation. In other words, when modification of the diffusion coefficient engendered by hydrodynamic interactions is taken into account, the above analysis predicts that no coagulation should occur at all because viscous friction forces become infinitely large as the particles approach each other. Thus, unless an attractive interaction is present between the particles, a colloidal suspension is inherently stable. To this end, one should recognize the importance of the London–van der Waals attraction forces in the coagulation process. Figure 12.5 shows the variation of the stability ratios in the absence of electrostatic repulsion for the case of a constant and a variable diffusion coefficient. With D being a constant, the stability ratio decreases with (AH /kB T ) and it has a value of unity at AH /kB T → 0 as was shown earlier in Figure 12.3. However, when correction for D is considered, the value of the stability ratio is higher than unity for AH /kB T ≤ 75 indicating the effects of hydrodynamic interactions in retarding the coagulation process. This is strictly due to the decrease of the particle mobility close to a rigid surface. The expression for U (h) used in Figure 12.5 is given by AH U (h) = − 6



2 h2 + 4h 2 + + ln h2 + 4h (h + 2)2 (h + 2)2



(12.59)

where h is the dimensionless gap width normalized using the particle radius, a. Experimental results for rapid coagulation confirm the analysis summarized by Eq. (12.57). Figure 12.6 and Table 12.2 show the stability ratio for rapid coagulation of polystyrene

Figure 12.5. Variation of stability ratio W with AH /kB T for constant and variable D. The London–van der Waals interaction potential is given by Eq. (12.59).

444

COAGULATION OF PARTICLES

Figure 12.6. Stability ratio for rapid coagulation of polystyrene in water as a function of the sphere radius. Solid line: prediction from retarded van der Waals potential. Symbols: data from Table 12.2. (Russel et al., 1989).

particles in water as a function of initial radius. The theoretical curve uses a retarded dispersion potential (Russel et al., 1989). Higashitani and Matsuno (1979) modified the population balance equation given by Eq. (12.20) to account for the interparticle potential and a variable diffusion coefficient. For the initial stage of the coagulation of a monodisperse system where ai and aj

TABLE 12.2. Data for Rapid Coagulation Experiments with Polystyrene Particles (Russel et al., 1989). 2a (µm)

W

AH /kB T

Mathews and Rhodes (1970)

0.714

1.83

1.5

Lips and Willis (1973)

0.207 0.357 0.500

1.78 1.89 1.82

1.9 1.2 1.7

Lichtenbelt et al. (1974)

0.091 0.109 0.176 0.234 0.357

1.67 1.82 1.59 2.00 1.96

2.7 1.5 4.1 0.7 1.7

Feke and Schowalter (1983)

0.675

1.87

1.1

12.5

BROWNIAN COAGULATION

445

are not too widely different, the modified population balance equation takes the form 1 dnk = dt 2W

i=k−1 

i+j =k;i=1

βij ni nj −

∞ 1  nk βik ni W i=1

k = 1, 2, 3, . . . , 10

(12.60)

where βij is defined by Eq. (12.36), W is defined by Eq. (12.57), and G(h) is given by Eq. (12.54). The experimental results of Higashitani and Matsuno (1979) for (nk /N0 ) variation with time are given in Figure 12.7 for polystyrene particles. The experimental conditions were such that the electrostatic repulsion forces were absent. Excellent agreement with theory is shown. This plot is similar to the case of the Smoluchowski Brownian motion coagulation shown in Figure 12.2 where W = 1 and the diffusion coefficient is constant. The dispersion potential was given by Eq. (12.59) with AH = 9 × 10−21 J. The excellent agreement with the solution of Eq. (12.60) is a further verification of the theoretical analysis presented for rapid coagulation. The dimensionless groups that control coagulation due to Brownian motion with added interparticle potential can be arrived at by making use of the dimensionless

Figure 12.7. Rapid Brownian flocculation for polystyrene spheres with a = 0.487 µm and W = 1.74. The calculated curves are from Eq. (12.60) (Higashitani and Matsuno, 1979).

446

COAGULATION OF PARTICLES

form of the flux or rate equation. The collision rate between the j particles and the i test particle, including transfer by convection (dropping the index j ), is given for steady state in Cartesian coordinates by  
n dU d dn − + ux n = 0 (12.61) D − dx dx kB T dx where ux is the fluid velocity in x-direction. Strictly speaking, ux is equal to the particle hydrodynamic velocity vx . The total interaction potential, U , appearing in Eq. (12.61) can be represented as a sum of the attractive van der Waals interaction, UA , and an electric double layer interaction, UR . The non-retarded van der Waals interaction potential is usually given by UA = AH f (X)

(12.62)

where AH is the Hamaker constant. The electric double layer interaction potential for identical charged spheres is given by UR = 2π ǫ ψs2 a g(X, κa) (12.63) The functions f and g are dimensionless functions and X=

x a

(12.64)

Let

ux (12.65) a γ˙ where a γ˙ is the characteristic velocity and γ˙ is the shear rate. Note that Ux is the dimensionless velocity and is different from the interaction potential, U . Making use of Eqs. (12.62) to (12.65), Eq. (12.61) becomes 
  df (X) n n dg(X, κa) dn d D − AH − 2π ǫψs2 a − + a γ˙ Ux n = 0 dX a dX kB T dX kB T dX Ux =

(12.66)

Assuming D to be a constant that is equal to 2D∞ and rearranging, Eq. (12.66) becomes  
 

d AH dn df (X) n 4π ǫψs2 a dg(X, κa) 1 a 2 γ˙ − − + Ux n = 0 − n dX dX kB T dX 2 kB T dX 2 D∞ (12.67)

Let AH kB T

(12.68)

4π ǫψs2 a kB T

(12.69)

NA = NR =

12.5

BROWNIAN COAGULATION

447

and Pe =

a 2 γ˙ D∞

(12.70)

where Pe is the Peclet number which is a measure of convection to diffusion transport, NA is a dimensionless parameter that describes the relative importance of the dispersion potential to Brownian motion, NR is a dimensionless parameter that describes the relative importance of the electrostatic potential to Brownian motion. Particle coagulation is governed by the dimensionless groups given by Eq. (12.71) in addition to the scaled inverse Debye length κa. The governing equation is   n Pe df dg dn d − nNA − NR + Ux n = 0 − dX dX dX 2 dX 2

(12.71)

It is possible to define other dimensionless groups. They are given by 6π a 3 γ˙ µ kB T

Pe =

(12.72)

= hydrodynamic potential/Brownian motion Nr =

4π ǫψs2 a NR = NA AH

(12.73)

= electrostatic potential/dispersion potential and Nf =

Pe 6π a 3 γ˙ µ = NA AH

(12.74)

= hydrodynamic potential/dispersion potential The dimensionless parameters of Eqs. (12.72), (12.73) and (12.74) represent one possible form of non-dimensionalization of the governing equations. Different research groups use different combinations of variables to obtain the dimensionless parameters. For instance, dimensionless parameters of the form CR =




Nr Nf



=

2ǫψs2 3a 2 µγ˙

(12.75)

= electrostatic force/hydrodynamic force and CA =

1 AH = 6Nf 36π µa 3 γ˙

= dispersion potential/hydrodynamic potential

(12.76)

448

COAGULATION OF PARTICLES

are also used in the literature. It should be recognized that additional dimensionless groups would enter into the analysis when retarded dispersion expressions, unequal sphere sizes, ion size, gravitational and inertial forces, surface tension, variable diffusion coefficient, and dielectric constant are considered. It is clear from Eq. (12.71) that by setting NA = NR = Pe = 0, the coagulation is due solely to Brownian motion. We turn our attention now to coagulation due to a non-zero Peclet number.

12.6

COAGULATION DUE TO SHEAR

12.6.1 The Smoluchowski Solution in the Absence of Brownian Motion Smoluchowski (1917) was the first to study particle coagulation due to shear. This represents the case of the Peclet number being very large (Pe → ∞). The simplest case is that of a laminar shear flow having a constant shear rate γ˙ . It is assumed that the particles follow straight streamlines. This situation is equivalent to the case of negligible inertia. The geometry of the flow is shown in Figure 12.8. For the purpose of the analysis, we consider a stationary test spherical particle of radius ai located at the origin with particles having a radius of aj moving towards it along the flow streamlines. When the centers of the two spheres are less or equal to (ai + aj ), coagulation occurs. Equivalently, contact between the two spheres occurs when y ≤ (ai + aj ) sin θ as shown in Figure 12.9. The velocity in x-direction ux is given by ux = γ˙ y

Figure 12.8. Coagulation due to shear for particles of radii ai and aj .

(12.77)

12.6

COAGULATION DUE TO SHEAR

449

Figure 12.9. Flow geometry for Smoluchowski coagulation under shear flow (adapted from Probstein, 2003).

where γ˙ is the constant shear rate. The number of aj particles per unit time entering element dy is given by 2[(ai + aj ) cos θ](γ˙ y) nj dy The total number of collisions per unit time between the j particles and the test particle is given by  ai +aj 2(2) [(ai + aj ) cos θ](γ˙ y)nj dy (12.78) 0

where the factor 2(2) in Eq. (12.78) accounts for the four quadrants of the test sphere. Substituting for y and including the number concentration of all the test particles, Eq. (12.78) yields 3

Jij = 4ni nj (ai + aj ) γ˙



π/2

cos2 θ sin θ dθ

0

Hence Jij =

4 (ai + aj )3 γ˙ ni nj 3

(12.79)

450

COAGULATION OF PARTICLES

with

4 (ai + aj )3 γ˙ (12.80) 3 where Jij is the collision frequency per unit volume. For a monodisperse system, at the initial stage of coagulation, one can assume that ai = aj = a, ni = nj = n, and Jij = J0,sh leading to βij =

J0,sh =




 32 a 3 γ˙ n2 3

and β =




 32 a 3 γ˙ 3

where J0,sh is Smoluchowski collision frequency per unit volume due solely to shear. The population balance Eq. (12.20) becomes ∞ i=k−1  dnk 1  32 3 32 3 = a γ˙ ni nj − nk a γ˙ n dt 2 i=1 3 3 i=1

(12.81)

Summing over all k, the above equation gives dNtot 16 2 = − a 3 γ˙ Ntot dt 3

(12.82)

where Ntot =

∞ 

ni

i=1

The volume fraction of the particles αp at the initial stage of coagulation is given by αp =

4 3 π a Ntot = constant 3

Making use of αp definition, Eq. (12.82) becomes dNtot 4 = − γ˙ αp Ntot dt π

(12.83)

The initial condition is given by Ntot (0) = N0

at t = 0

The solution of Eq. (12.83) is
 t Ntot (t) = N0 exp − tsh where tsh =

π 4γ˙ αp

(12.84)

(12.85)

12.6

COAGULATION DUE TO SHEAR

451

tsh is a characteristic coagulation time for coagulation due solely to shear. Comparing Eqs. (12.40) and (12.82) leads to 4a 3 γ˙ µ rate of coagulation due to shear = rate of coagulation due to Brownian motion kB T One can observe that the above ratio increases with increasing particle radius, shear rate, and fluid viscosity. For a = 10−5 m (10 µm), γ˙ = 1s−1 , µ = 10−3 Pa s, and T = 300 K, the ratio of shear coagulation to diffusion becomes about 1000. Reducing the particle size to one micron, the ratio becomes about unity. Coagulation in the absence of the Brownian motion represents the case of a very high Peclet number (Pe → ∞). 12.6.2 Coagulation due to Shear in the Absence of Brownian Motion: With Hydrodynamic and Field Forces In the previous section, we dealt with particle coagulation where the flow field is unaffected by the presence of the particles. Here we shall deal with the case of coagulation due to shear with the full description of the flow field. Simple shear will be assumed where ux = γ˙ y. The simplest case of laminar shear flow past a sphere is given by Cox et al. (1968). The streamlines of the flow are shown in Figure 12.10. The streamlines can be thought of as the tracing of the trajectory of a neutral particle with a zero radius. Figure 12.10(b) clearly shows that there exist open and closed streamlines separated by a limiting streamline or surface. The case of two spheres in a simple shear flow, the creeping flow problem was solved by Lin et al. (1970), Batchelor and Green (1972), Kao et al. (1977), and Kim and Mifflin (1985). With the origin of the coordinates placed at the center of sphere 1,

Figure 12.10. Laminar shear flow about a sphere: (a) flow field far from sphere; (b) equatorial streamlines around a sphere (x-y) plane (Cox et al., 1968).

452

COAGULATION OF PARTICLES

Figure 12.11. Coordinates for trajectory equations for two spheres in a simple shear flow.

the relative position of sphere 2 is given by the spherical coordinates (r, θ, φ) as shown in Figure 12.11. For a simple shear far from the spheres, the flow velocities are given by ux = γ˙ y

(12.86)

uy = uz = 0

(12.87)

with The relative trajectories of sphere 2 with respect to sphere 1 are given by dr = γ˙ r(1 − A) sin2 θ sin φ cos φ dt

(12.88)

dθ = γ˙ (1 − B) sin θ cos θ sin φ cos φ dt

(12.89)

and    dφ 1  2 2 2 = −γ˙ sin φ + B cos φ − sin φ dt 2

(12.90)

where t is time. Coefficients A and B are functions of radial distance r and size ratio a1 /a2 . They are tabulated in Table 12.3 for equal-sized spheres (a1 = a2 ) with R (= r/a), where r is the separation distance between the spheres centers and a is the radius of the equal-sized spheres 1 and 2. A and B assume different values for unequal spheres. Figure 12.12 shows the equatorial trajectories for two spheres of radii a1 and a2 . It is clear that the trajectories are similar to the case around a sphere in simple shear as is shown in Figure 12.10. The flow is characterized by open and closed trajectories. In the absence of the Brownian motion and external forces, two spheres approaching each other from infinity are unable to penetrate the region of closed trajectories and

12.6

COAGULATION DUE TO SHEAR

453

TABLE 12.3. Values of A, B, and C for Equal Sized Spheres. (Taken from Zeichner and Schowalter, 1977.) R 2.0000 2.0001 2.0025 2.0100 2.0401 2.0907 2.1621 2.2553 2.3709 2.5103 2.6750 2.8662 3.0862 3.3370 3.6213 4.7048 6.2149 11.1139 20.1353

A

B

C

1.0000 0.9996 0.9900 0.9619 0.8679 0.7505 0.6313 0.5214 0.4248 0.3424 0.2735 0.2167 0.1704 0.1331 0.1033 0.0468 0.0204 0.0036 0.0006

0.4060 0.3213 0.2762 0.2461 0.1996 0.1608 0.1275 0.0988 0.0748 0.0553 0.0399 0.0281 0.0193 0.0130 0.0086 0.0023 0.0006 0.0000 0.0000

0.0000 0.0004 0.0098 0.0374 0.1310 0.2510 0.3777 0.5017 0.6190 0.7287 0.8306 0.9254 1.0133 1.0950 1.1709 1.3666 1.5207 1.7310 1.8512

Figure 12.12. Equatorial trajectories of two spheres in a simple shear (schematic). The solid lines are the relative trajectories of a sphere of radius a2 with respect to a reference sphere of radius a1 . Two kinds of trajectories exist: separating (or open) and closed ones, separated by a limiting trajectory. The shaded area is the region of closed trajectories (van de Ven, 1989).

454

COAGULATION OF PARTICLES

hence they are unable to approach each other closer than distance dmin where dmin is the minimum distance of approach of two spheres on the limiting trajectory (van de Ven, 1989). For equal-sized spheres, dmin /a1 = 4.2 × 10−5 (Arp and Mason, 1977) and for a2 /a1 = 0, i.e., a single sphere, dmin /a1 = 0.16 (Cox et al., 1968). It becomes clear, then, that when the hydrodynamic interactions are taken into consideration, no coagulation takes place between two particles under a simple shear flow in the absence of the Brownian motion and van der Waals dispersion (attractive) forces. This is in contrast to Smoluchowski’s analysis. In order for coagulation to occur under a simple shear flow, an attractive force needs to be present between the particles. In the above discussion, inertial effects are assumed to be negligible. We consider now the case of coagulation of two equal-sized spheres in laminar shear flow under the influence of interaction forces. Equation (12.88) can be regarded as a force balance along the line between sphere centers where the hydrodynamic force acts on the spheres; i.e.,
 dr Force = 6π aµ dt For the case of a non-zero interaction force, Eq. (12.88) becomes 6π aµ

  dr ´ sin2 θ sin φ cos φ = 6π aµ γ˙ r(1 − A) dt  + C(r/a) interaction forces in r-direction

(12.91)

The implicit assumption made in writing Eq. (12.91) is that the interaction forces act only along the line of centers and are balanced by the component of the hydrodynamic force in that direction, so that the net force on each sphere is zero and there is no additional torque (van de Ven and Mason, 1976), Function C(r/a) is a correction parameter to Stokes law due to the presence of a second sphere. When the interaction forces are due to the London–van der Waals (attractive) and to electrostatic (repulsive) forces, one can write The London–van der Waals force is given by FA (r) = −

dUA dr

(12.92)

where for the case of unretarded dispersion potential UA between two equal-sized spheres, it can be given by AH UA = − 6




2  2 R −4 2 + + ln R2 − 4 R2 R2

with R = r/a. In dimensionless form, Eq. (12.92) becomes fA (R) =

a 1 dUA FA (r) = − AH AH dR

(12.93)

12.6

COAGULATION DUE TO SHEAR

455

The electrostatic repulsive force between the two spheres is given by FR (r) = −

dUR dr

(12.94)

where for a constant low surface potential, the electrostatic potential UR for two equal-sized spheres is given by Eq. (5.142) as UR = 2π ǫaψs2 ln[1 + exp(−κa(R − 2))] In dimensionless form, Eq. (12.94) becomes fR = −

1 dUR 4π ǫaψs2 dR

(12.95)

leading to fR =

1 FR (R) 4π ǫψs2

Making use of Eqs. (12.93) and (12.95), Eq. (12.91) becomes   C(R) AH dr 2 2 = [γ˙ r(1 − A) sin θ sin φ cos φ] + fA (R) 4π ǫψs fR (R) + dt 6π aµ a (12.96) Non-dimensionalizing the time t using γ˙ as τ = t γ˙ and making use of Eqs. (12.73) and (12.74), Eq. (12.96) becomes in dimensionless form dR C(R) = R(1 − A) sin2 θ sin φ cos φ + (Nr fR + fA ) (12.97) dτ Nf The corresponding trajectory equation represented by Eqs. (12.89) and (12.90) becomes dθ = (1 − B) sin θ cos θ sin φ cos φ (12.98) dτ and   dφ 1 2 2 2 = − sin φ + B(cos φ − sin φ) dτ 2

(12.99)

The function C(R) is related to the parameter G(h) given by Eq. (12.54) for the case of diffusion coefficient, where G(R) ∼ =

2 C(R)

(12.100)

456

COAGULATION OF PARTICLES

and C(R) ∼ =

4(3R 2 − 10R + 8) 6R 2 − 11R

(12.101)

where R = r/a = 2 + h. Tabulation of C(R) is given in Table 12.3. Various expressions for A, B, and C are given by Higashitani et al. (1982) and van de Ven and Mason (1976) for R → 2 and R → ∞. Solution of the trajectory equation provides the trajectories of the colliding spheres. For the case of Nr = 0 (i.e., absence of a repulsive force) and Nf = 1, Feke and Schowalter (1983) provide the trajectories at z = 0 plane (equatorial plane) as shown in Figure 12.13. There are three types of trajectories. The first type is an open trajectory initiated from downstream and extending to upstream infinity. Here, the two spheres do not coagulate. The second type of trajectory leads to coagulation. It crosses the x-axis and reverses direction prior to collision. This is shown by the shaded area of Figure 12.13. The third type of trajectory leads to coagulation before crossing the x-axis which is the plane of shear. Takamura et al. (1981) visualized the trajectories of shear-induced collisions between two equal-sized latex spheres for three cases: (a) repulsion is dominant, (b) weak attraction, and (c) strong attraction. Figure 12.14 shows the three cases. The two curves in each figure represent the projection on the xy-plane of the paths of the centers of the two spheres. At the origin is the exclusion sphere which cannot be penetrated by either colliding sphere. Figure 12.14(a) demonstrates the effect of a net repulsive force on the spheres trajectories. The spheres initially approach each other

Figure 12.13. Trajectories in the z = 0 plane for two spheres in a simple shear flow, with Nr = 0 and Nf = 1: (1) open trajectories extending from upstream to downstream infinity; (2) trajectories leading to collision after crossing the plane of shear (y = 0) and reversing direction; (3) trajectories leading to collision prior to crossing the plane of shear (Feke and Schowalter, 1983).

12.6

COAGULATION DUE TO SHEAR

457

Figure 12.14. Trajectories of shear induced collisions of 2.6 µm PS latex spheres in 50% aqueous glycerol showing the projection on the xy-plane of the paths of the sphere centers from the midpoint between them. At the center is the exclusion sphere which cannot be penetrated when the collision occurs in the xy-plane (Takamura et al., 1981).

with their centers ±0.15a off the x-axis. However, after they encounter each other, they recede with their centers ±0.8a away from the x-axis. Figure 12.14(b) shows the case of a weak attraction upon increasing the KCl electrolyte concentration to 10 mM. The paths of approach and recession of the two spheres become almost a mirror image of each other. By further increasing the electrolyte concentration to 100 mM, the net force becomes attractive and a doublet is formed upon the spheres collision as is shown in Figure 12.14(c).

458

COAGULATION OF PARTICLES

Figure 12.15. Trajectory of real systems (Higashitani et al., 1982).

To evaluate the coagulation rate, it is necessary to know the limiting trajectory within which all particles collide. A limiting trajectory for the case of two equal-sized spheres is defined as that which terminates at φ = 0 with r = 2a (see Figure 12.11). The area occupied by the limiting trajectories is usually evaluated by retracing backward in time the path of the particles. The limiting interception area is shown in Figure 12.15 by the shaded area. The corresponding limiting interception area for Smoluchowki’s analysis is shown for comparison in Figure 12.16. The rate of coagulation, J , between particles of equal-sized spheres per unit volume is given by counting the number of particles crossing the interception area, where J = 4n2 γ˙



y0

yz(y)dy

(12.102)

0

Figure 12.16. Trajectory of Smoluchowski model (Higashitani et al., 1982).

12.6

COAGULATION DUE TO SHEAR

459

where z = z(y) is the function defining the boundary of the interception area and y0 is the y value of the boundary at z = 0. Recalling Smoluchowski’s collision rate J0,sh defined as
 32 J0,sh = a 3 γ˙ n2 (12.103) 3 it is possible to define a stability ratio W as W =

J0,sh J

(12.104)

Large W values indicate a stable colloidal system. In some literature, the inverse of W is used, and W −1 is called the shear coagulation coefficient. Figure 12.17 shows the stability ratio W as a function of Nf for the case of Nr = 0. At small Nf values, W becomes less than unity, indicating a strong attractive London–van der Waals force. However, at large value of Nf , where the London–van der Waals force becomes weaker relative to the hydrodynamic force, W increases rapidly indicating a high hydrodynamic resistance as compared to London–van der Waals forces for the approaching particles and a reduction in J . The preceding interpretation of Figure 12.17 is to keep the hydrodynamic potential 6π a 3 γµ ˙ constant (see Eq. 12.74) and the change in Nf is due to the Hamaker constant. The variation in AH reflects changes in the attractive London–van der Waals forces. One can also use Figure 12.17 to examine the effect of increasing the flow strength γ˙ . Here AH is kept constant. Recognizing that γ˙ appears both in W and in Nf , a change in Nf from, say 102 to 105 gives a corresponding change in W from ∼5 to ∼20. This

Figure 12.17. Comparison of the stability ratios, W = 32a 3 γ˙ n2 /(3J ), for flow-induced coagulation for Pe → ∞ using unretarded potential. Dashed line: trajectory analysis of Zeichner and Schowalter (1977). Solid line: Feke and Schowalter (1983).

460

COAGULATION OF PARTICLES

means that J at Nf = 105 is ∼250 times that at Nf = 102 . Consequently, increasing γ˙ by a factor of 103 has a profound effect on increasing the collision rate per unit volume. Now let us study the effect of viscosity by keeping other parameters constant. Nf contains the viscosity µ and W does not implicitly contain µ. Increasing Nf from 102 to 105 would give a corresponding change in J of 5/20 = 0.25. This means that increasing the fluid viscosity by a factor of 103 only reduces J by a factor of 4. The integration of the trajectory Eqs. (12.97) to (12.99) for different Nr and Nf produces stability diagrams (van de Ven and Mason, 1976; Zeichner and Schowalter, 1977). A typical stability plot is shown in Figure 12.18. It should be recalled that for a given fluid viscosity, particle radius, medium dielectric constant, and Hamaker constant, one obtains Nr ∝ ψs2 ∝ (repulsion force) and Nf ∝ γ˙ ∝ (flow strength). √ Consequently, Nr represents changes in the surface potential ψs and Nf represents the strength of the flow field. From an experimenter’s point of view, ψs is controlled by the ionic strength and pH of the electrolyte solution. The stability diagram shown in Figure 12.18 is divided into three regions. In region 1, coagulation occurs in the primary minimum. In region 2, coagulation occurs in the secondary minimum, and finally, in region 3, no coagulation occurs (Zeichner and Schowalter, 1977). For example, if one is to start with a sol (a colloidal suspension) under no shear containing spherical particles with, say, Nr = 20, then upon subjecting the sol to increasing shear, doublets coagulating in the primary minimum would be formed. Increasing Nf beyond 5 × 103 would cause the doublet to deflocculate. For Nr = 100, if a sol is sheared then doublets would form at the secondary minimum. A further increase in the rate of shear can lead to deflocculation of the doublets out of

Figure 12.18. Stability plane for shear flow (Zeichner and Schowalter, 1977). Region 1 – primary minimum coagulation; Region 2 – secondary minimum coagulation; Region 3 – stable. (Calculated for unretarded attraction, constant surface potential, κa = 100, and dmin /a = 2.004.)

12.6

COAGULATION DUE TO SHEAR

461

Figure 12.19. Stability diagram for doublet formation in a simple shear flow at Pe → ∞, with a constant-potential boundary condition and the non-retarded Hamaker form for the attraction force (Zeichner and Schowalter, 1977).

the secondary minimum. Increasing the shear rate further would cause the particles to coagulate into the primary minimum. At still higher shear rates, doublets can deflocculate out of the primary minimum. If one is to decrease the shear rate, the dispersed sol will again coagulate into the primary minimum. Further lowering of the shear rate will not, however, cause deflocculation or a shift to secondary minimum flocculation. The boundaries of the various regions shift with κa as shown in Figure 12.19. This type of a stability diagram is very important in the formation of emulsions and in the process of demulsification under shear. In all cases discussed above, the Brownian motion was not considered and we dealt with cases corresponding to large Pe. Figure 12.18 is useful in interpreting many industrial processes where colloidal systems are of concern. For example, Yan and Masliyah (1993) found that for solids stabilized oil-in-water emulsions, it is possible to demulsify such emulsions by the addition of fresh oil under mixing action. The mixing action breaks down the added fresh oil into large oil droplets which act as scavengers to the solids-coated oil emulsion droplets. Here coagulation occurs in the primary minimum and hence scavenging takes place. In this process, the mixing action, i.e., the shear rate, is very important. Very high mixing rates were found to be detrimental to the scavenging process because the added fresh oil would form stable emulsion at large Nf values. Another example is in the processing of oil sands and the recovery of bitumen. Oil sands containing bitumen are diluted with hot water in the presence of NaOH and are slurried in a rotational conditioning drum. After the liberation of the bitumen from the sand grains, it becomes desirable to avoid the coagulation of the liberated bitumen with the fines (or clays) in the slurry. Proper Nr and Nf values should be maintained so that the slurrying operation occurs in region 3. This depends on the amount of NaOH added and the degree of mixing in the conditioning equipment.

462

COAGULATION OF PARTICLES

Figure 12.20. Rate of doublet formation for spheres as a function of Pe, showing the results from the perturbation expansions in the Pe ≪ 1 and Pe ≫ 1 limits for AH /kB T = 2 (solid lines) and 20 (dashed lines). These results were obtained in absence of any repulsive force (Nr = 0) (Feke and Schowalter, 1985). The Peclet number in this figure is given as 3π a 2 γµ/k ˙ BT .

For cases of intermediate Pe values, the combined effects of the Brownian diffusion on shear-induced coagulation of colloidal dispersions are dealt with by Feke and Schowalter (1985). The protocol is to combine the analysis made for the Brownian and shear coagulation. Figure 12.20 shows the rate of doublet formation normalized with Jo,Br . Two regions are given, one for Pe ≪ 1 and Pe ≫ 1. No solution was provided for the intermediate region. Figure 12.20 indicates that in the region of Pe ≪ 1, increasing γ˙ (all other parameters being constant) has little effect on the collision rate per unit volume J . For the case of Pe ≫ 1, J increases rapidly with changes in the flow strength. The analysis of the preceding sections clearly indicates the complexity of the particle-particle coagulation process where the Brownian motion, hydrodynamic forces and interaction forces due to electrostatic and dispersion can play a major role. Experimental studies tend to confirm the theoretical analysis for rapid coagulation. However, deviation from theory occurs for slow coagulation where appreciable repulsive forces are present (Zeichner and Schowalter, 1979; Kihira et al., 1992).

12.7

NOMENCLATURE

A, B, C a, a1 , a2 , a3 AH CA

flow functions, dimensionless particle radius, m Hamaker constant, J dimensionless parameter describing the relative importance of dispersion and hydrodynamic potentials (for forces), AH /36π µa 3 γ˙

12.7

CR d d⊥ D = D D∞ Di , Dj f fA (R) fR (R) f (x) F (r) g(X, κa) h i, j, k Jij J0,Br J0,sh j∗∗ D kB K m n n∞ ni , nj ni∞ , nj ∞ NA Nf

Ni N0 Nr NR

NOMENCLATURE

463

dimensionless parameter describing the relative importance of electrostatic and hydrodynamic forces, 2ǫψs2 /3a 2 µγ˙ correction factor for the diffusion coefficient for motion parallel to the surface of the spheres, dimensionless correction factor for the diffusion coefficient for motion normal to the surface of the spheres, dimensionless particle diffusion coefficient, m2 /s particle diffusion coefficient tensor, m2 /s particle diffusion coefficient at infinite dilution, m2 /s diffusion coefficient of particles i and j , respectively, m2 /s friction coefficient, Eq. (12.7), Ns/m dimensionless dispersion potential dimensionless electrostatic potential dimensionless function for the dispersion potential, Eq. (12.62) interaction force function, N dimensionless function for the electrostatic potential dimensionless gap between two surfaces, h = R − 2 particle species collision rate (frequency) per unit volume, m−3 s−1 Smoluchowski collision rate per unit volume due to Brownian diffusion, 8kB T n2 /3µ, m−3 s−1 Smoluchowski collision rate per unit volume due to simple shear, (32/3)a 3 γ˙ n2 , m−3 s−1 diffusive flux based on number concentration, (m−2 s −1 ) Boltzmann constant, J/K function defined by Eq. (12.38), m3 /s mass of a particle, kg number of particle per unit volume (number concentration), m−3 number concentration of particle at time zero, m−3 number concentration of particle species i and j , respectively, m−3 number concentration of particle species i and j in the undisturbed state, respectively, m−3 dimensionless parameter describing the relative importance of the dispersion potential and Brownian motion, AH /kB T dimensionless parameter describing the relative importance of the hydrodynamic and dispersion potentials (or forces), 6π a 3 γµ/A ˙ H total number of released particles per unit area at time zero and x = 0, m−2 value of Ntot at time zero, m−3 dimensionless parameter describing the relative importance of electrostatic and dispersion potentials (or forces), 4π ǫaψs2 /AH dimensionless parameter describing the relative importance of the electrostatic potential and Brownian motion, 4π ǫaψs2 /kB T

464

COAGULATION OF PARTICLES

Ntot Pe r R t tBr tsh T ux Ux U UA UR v vi , vj , vk W x X

total number of particles per unit volume (number of concentrations), m−3 Peclet number, a measure of convection to diffusional transport, dimensionless radial position; dimensionless center-to-center distance between two spheres, m dimensionless center-to-center distance between two spheres, r/a time, s characteristic half-time for coagulation due to Brownian diffusion, π µa 3 /αp kB T , s characteristic half-time for coagulation due to shear, π/4γ˙ αp , s absolute temperature, K fluid velocity in x-direction, m/s dimensionless fluid velocity ux /(a γ˙ ) interaction potential, J dispersion potential, normally assumed to be attractive, J electrostatic potential, normally assumed to be repulsive, J particle velocity, m/s volume of particle species i, j and k, respectively, m3 stability ratio, dimensionless Cartesian coordinate, m dimensionless distance or coordinate, x/a

Greek Symbols αp β βij γ˙ ǫ θ κ µ τ φ ψs

12.8

volume fraction of particles, dimensionless function relating t , Eq. (12.8); collision frequency function, m3 /s collision frequency function pertaining to particles i and j , m3 /s shear rate, s−1 dielectric permittivity of a medium, C/Vm spherical coordinate inverse Debye length, m−1 fluid viscosity, Pa s dimensionless time, t γ˙ spherical coordinate surface potential, V

PROBLEMS

12.1. We have discussed Brownian coagulation under a field force in terms of the stability ratio, W . Let us investigate the stability ratio, defined by Eq. (12.57), where van der Waals and repulsive forces are considered for the case of monosized particles and variable diffusion coefficient. The van der Waals interaction potential can be taken from Eq. (12.59) and the electrostatic potential from

12.8

PROBLEMS

465

Eq. (5.164) with a1 = a2 = a. Assume a solution temperature of 20◦ C and a Hamaker constant of 10−21 J. Evaluate the stability ratio and comment on the sensitivity of W to changes in the system properties for the following cases: Case 1: Uncharged spheres in a mono-sized particle dispersion. Case 2: Particle surface zeta potential of 25 mV in a 0.001 M (2 : 1) electrolyte solution. Calculate the stability ratio for three different particle sizes in the monodisperse suspension, namely a = 0.1, 1, and 10 µm. Case 3: Particle radius, a = 1.0 µm in a 0.001 M (2 : 1) electrolyte solution. Calculate W for particle surface zeta potentials of 10 and 50 mV. Also, plot the stability ratio, W against zeta potential for the range of 5 to 50 mV. Case 4: Particle radius of 1.0 µm and particle surface zeta potential of 25 mV. Calculate the stability ratio in a 0.001 M electrolyte solution for a (1 : 1), (2 : 1), and (3 : 1) electrolyte. 12.2. We have discussed coagulation due to Brownian motion and due to shear using the Smoluchowski approach. Assume additivity of the coagulation rates due to these two mechanisms (see Eqs. 12.40 and 12.83). Show that the rate of change of total particle concentration is given by 4γ˙ αp dNtot 4kB T 2 =− Ntot − Ntot dt 3µ π where Ntot is the total number of particles per unit volume, m−3 . The other symbols have their usual definition. (a) Show that for an initial particle concentration, Ntot (0) = N0 , the solution to the above coagulation rate equation is given by R exp(−4γ˙ αp t/π ) Ntot = N0 1 + R − exp(−4γ˙ αp t/π )

(12.105)

where R=

3γ˙ αp µ π kB T N0

The parameter R can be thought of as being a measure of shear to Brownian motion. (b) Show that in the limiting case of γ˙ → 0 (which makes R → 0), Eq. (12.105) degenerates to Eq. (12.41) for the case of Brownian coagulation. (c) Show that in the limiting case of γ˙ → ∞, making R → ∞, Eq. (12.105) becomes identical to Eq. (12.84) for the case of coagulation due to shear. (d) Plot the variation of Ntot /N0 with time for µ = 0.001 Pa s, T = 300 K, N0 = 1014 m−3 and αp = 0.1. Use γ˙ = 0.1, 1, 3, 10, and 100 s−1 . Comment on your plot.

466

12.9

COAGULATION OF PARTICLES

REFERENCES

Arp, P. A., and Mason, S. G., The kinetics of flowing dispersion. VIII. Doublets of rigid spheres (theoretical), J. Colloid Interface Sci., 61, 21–43, (1977). Batchelor, G. K., and Green, J. T., The hydrodynamic interaction of two small freely-moving spheres in a linear flow field, J. Fluid Mech., 56, 275–400, (1972). Brenner, H., The slow motion of a sphere through a viscous fluid towards a plane surface, Chem. Eng. Sci., 16, 242–251, (1961). Cox, R. G., Zia, I. Y. Z., and Mason, S. G., Particle motions in sheared suspensions. XXV. Streamlines around cylinders and spheres, J. Colloid Interface Sci., 27, 7–18, (1941). Derjaguin, B. V., and Landau, L., Theory of the stability of strongly charged lyophobic sols and the adhesion of strongly charged particles in solutions of electrolytes, Acta Physicochim. URSS, 14, 633–662, (1941). Derjaguin, B. V., and Muller, V. M., Slow coagulation of hydrophobic colloids, Dokl. Akad. Nauk. SSSR, 176, 738–741, (1967). Feke, D. L., and Schowalter, W. R., The effect of Brownian diffusion on shear-induced coagulation of colloidal dispersion, J. Fluid Mech., 133, 17–35, (1983). Feke, D. L., and Schowalter, W. R., The influence of Brownian diffusion on binary flow-induced collision rates in colloidal dispersion, J. Colloid Interface Sci., 106, 203–214, (1985). Friedlander, S. K., Smoke, Dust and Haze: Fundamentals of Aerosol Dynamics, 2nd ed., Oxford University Press, New York, (2000). Fuchs, N., Über die stabilität und aufladung der aerosole, Z. Physik., 89, 736–743, (1934). Kao, S. V., Cox, R. G., and Mason, S. G., Streamlines around single spheres and trajectories of pairs of spheres in two-dimensional creeping flow, Chem. Eng. Sci., 32, 1505–1515, (1977). Kihira, H., Ryde, N., and Matijevic, E., Kinetics of heterocoagulation i. a comparison of theory and experiment, Colloids and Surfaces, 64, 317–324, (1992). Kim, S., and Mifflin, R. T., The resistance and mobility functions of two equal spheres in low Reynolds number flow, Phys. Fluids, 28, 2033–2045, (1985). Higashitani, K., and Matsuno, Y., Rapid Brownian coagulation of colloidal dispersions, J. Chem. Eng. Japan, 12, 460–465, (1979). Higashitani, K., Ogawa, R., Hosokawa, G., and Matsuno,Y., J. Chem. Eng. Japan, 15, 299–304, (1982). Honig, E. P., Roebersen, G. J., and Wiersema, P. H., Effect of hydrodynamic interaction on the coagulation rate of hydrophobic colloids, J. Colloid Interface Sci., 36, 97–109, (1971). Lichtenbelt, J. W. Th., Pathmanamoharan, C., and Wiersema, P. H., Rapid coagulation of polystyrene latex in a stopped flow spectrophotometer, J. Colloid Interface Sci., 49, 281–285, (1974). Lin, C. J., Lee, K. J., and Sather, N. F., Slow motion of two spheres in a shear field, J. Fluid Mech., 43, 35–47, (1970). Lips, A., and Willis, W. E., Low angle scattering technique for the study of coagulation, J. Chem. Soc. Faraday Trans. I, 69, 1226–1236, (1973). Mathews, B. A., and Rhodes, C. T., Studies of the coagulation kinetics of mixed suspensions, J. Colloid Interface Sci., 32, 332–338, (1970). Probstein, R. F., Physicochemical Hydrodynamics, An Introduction, 2nd ed., Wiley Interscience, New York, (2003).

12.9

REFERENCES

467

Russel, W. B., Saville, D. A., and Schowalter, W. R., Colloidal Dispersions, Cambridge University Press, Cambridge, (1989). Smoluchowski, M. von, Versuch einer mathematischen theorie der koagulationskinetic kolloider Lösungen, Z. Phys. Chem., 92, 129–168, (1917). Spielman, L. A., Viscous interactions in Brownian coagulation, J. Colloid Interface Sci., 33, 562–571, (1970). Spielman, L. A., Particle capture from low-speed laminar flows, Ann. Rev. Fluid Mech., 9, 297–319, (1977). Takamura, K., Goldsmith, H. L., and Mason, S. G., The microrheology of colloidal dispersions xii. trajectories of orthokinetic pair-collisions of latex spheres in a simple electrolyte, J. Colloid Interface Sci., 82, 175–189, (1981). van de Ven, T. G. M., Colloidal Hydrodynamics, Academic Press, London, (1989). van de Ven, T. G. M., and Mason, S. G., The microrheology of colloidal dispersion. IV. Pairs of interacting spheres in a shear flow, J. Colloid Interface Sci., 57, 505–516, (1976). Verwey, E. J. W., and Overbeek, J. Th. G., Theory of the Stability of Lyophobic Colloids, Elsevier, Amsterdam, (1948). Yan, Y., and Masliyah, J. H., Solids-stabilized oil-in-water emulsion: scavenging of emulsion droplets by fresh oil addition, Colloids Surf., 75, 123–132, (1993). Zeichner, G. R., and Schowalter, W. R., Use of trajectory analysis to study stability of colloidal dispersions in flow fields, AIChE J., 23, 243–254, (1977). Zeichner, G. R., and Schowalter, W. R., Effects of hydrodynamic and colloidal forces on the coagulation of dispersions, J. Colloid Interface Sci., 71, 237–253, (1979).

CHAPTER 13

DEPOSITION OF COLLOIDAL PARTICLES

13.1

INTRODUCTION

In Chapter 12 we discussed particle–particle interaction under Brownian motion and field forces. A stability ratio W was defined to assess the stability of a colloidal system. The cases considered were concerned with particles of similar size and consequently the analysis was of interest in studying the stability of colloidal systems under Brownian motion, shear and interparticle forces such as the London–van der Waals attractive and electrostatic repulsive forces. In this chapter we shall deal with the interactions between a small particle and a large surface, generally referred to as the collector. The small particle is of colloidal size that is much smaller than the collector with which it is interacting. The large surface can be a cylindrical fiber in a filter mat, a granular particle in a packed bed or simply a surface surrounding the colloidal dispersion, e.g., a container’s wall. Various deposition configurations are shown in Figure 13.1. We shall deal only with the process of deposition rather than with the process of detachment. The latter case is more complicated to analyze as the depth of the energy minimum in which a particle is captured is usually not known. However, the study of detachment itself can provide information regarding the strength of the forces between a particle and a surface (van de Ven, 1989; Liu et al., 2004). There are two approaches to the study of particle deposition. The first method is the Eulerian approach where the distribution of particles is evaluated in space. In the absence of colloidal forces and for infinitesimal particle size, the Eulerian approach Electrokinetic and Colloid Transport Phenomena, by Jacob H. Masliyah and Subir Bhattacharjee Copyright © 2006 John Wiley & Sons, Inc.

469

470

DEPOSITION OF COLLOIDAL PARTICLES

Figure 13.1. Various deposition geometries: (a) rotating disc; (b) impinging jet; (c) flow over a spherical collector; (d) cross flow over a cylinder; (e) flow along a flat plate; (f) sphere in shear flow; (g) flow parallel to a cylinder (Adamczyk, 1989a,b).

becomes that of the study of heat or mass transfer from submerged objects in a flowing stream. To this end, the generalized convection-diffusion equation is solved for the diffusing colloidal particles subject of the appropriate boundary conditions. The second method is the Lagrangian approach. Here attention is focused on a single particle trajectory which is described by Newton’s laws of motion. The particle path is followed and the particle trajectories determine the collision or capture efficiency between the colloidal particle and the larger collector particle. The forces associated with Newton’s law are deterministic in nature and consequently most of the literature studies employing the Lagrangian approach usually do not consider the Brownian motion and, hence, the diffusion of the particles. However, it is possible to include

13.2

CLASSICAL DEPOSITION MECHANISMS

471

the Brownian motion in the analysis and both approaches should, in principle, yield similar results (van de Ven, 1989). 13.2

CLASSICAL DEPOSITION MECHANISMS

Deposition of particles onto surfaces occurs due to several interparticle forces. However, in limiting cases it is possible to distinguish between the various modes of particle deposition. The various modes of deposition can be classified as follows: 13.2.1 Brownian Diffusion: Classical Convection–Diffusion Transport Submicron particles undergo Brownian motion, which enhances their deposition during flow past the surface of a collector. The governing transport equation is given by the convection–diffusion equation as u · ∇c = D∞ ∇ 2 c

(13.1)

where u is the fluid velocity, c is the particle concentration and D∞ is the particle diffusion coefficient given by the Stokes–Einstein equation as D∞ =

kB T 6π µa

(13.2)

This analysis treats the particles as diffusing non-interacting points with D∞ being a constant. The particle hydrodynamic velocity is the same as the fluid velocity. The solution of the Navier–Stokes equation provides the fluid velocity u. The mass transfer rate is related to the flow Reynolds and Schmidt numbers. In the limit of low Reynolds number (creeping) flow, the mass transfer is related to the flow Peclet number, P e, defined as LU∞ /D∞ where L and U∞ represent a characteristic length and the undisturbed fluid velocity, respectively. In this context, L represents the collector length scale.1 The Peclet number is obtained as the product of the Reynolds number (Re = LU∞ ρ/µ) and the Schmidt number (Sc = µ/(ρD∞ )), where ρ is the fluid density and µ is the fluid dynamic viscosity. The Peclet number represents the ratio of the convective force to the diffusive (Brownian) force. In general, for P e ≪ 1, the transport is dominated by diffusion, while for P e ≫ 1, the transport is dominated by convection. For small Peclet numbers, P e ≪ 1, the dimensionless mass transfer group as presented by the Sherwood number is directly proportional to P e. On the other hand, when P e ≫ 1, the Sherwood number becomes proportional to P e1/3 . 13.2.2 Interception Deposition Capture by interception assumes that the particles have a finite size and they are noninteracting and non-diffusing. The center of a particle follows exactly the undisturbed 1

In Chapter 12, the characteristic length scale was that of the colloidal particle radius, a.

472

DEPOSITION OF COLLOIDAL PARTICLES

Figure 13.2. Capture due to interception by a spherical collector.

fluid streamline past the collector. When the particle touches the collector, capture or deposition takes place. Capture by pure interception ignores the increased hydrodynamic resistance between the particles and the collector upon approach. Figure 13.2 shows capture by interception of a spherical particle by a spherical collector. Particles located above the limiting trajectory or streamline, ψL , will not intercept the collector. Particles located below the limiting trajectory will intercept the collector. As the particles follow the flow streamline, it is possible to evaluate the flux of particles that intercept a collector from the knowledge of the flow stream function. For the case of an isolated spherical collector, the flow stream function, under creeping flow conditions, is given by ψ=


 1 a3 3 U∞ r 2 − ac r + c sin2 θ 2 2 2r

(13.3)

The spherical coordinates are shown in Figure 13.2 together with the streamlines which are the loci of a constant ψ-value. The volumetric flow rate between any streamline and the collector axis is given, by definition, as 2π ψ. A particle is considered to be captured or intercepted by the collector when its center is within one particle radius from the collector surface. Consequently, all particles within such a limiting streamline are captured. The limiting streamline, L , for the special case of interception deposition is given by setting r = ap + ac and θ = π/2; hence ψL =

 
1 a3 3 1 U∞ (ap + ac )2 − ac (ap + ac ) + c 2 2 2 ap + ac

(13.4)

The total flux of the captured particles is given by 2π ψL n∞ . Capture efficiency is defined by the ratio of the actual captured particles to the idealized capture given by the area swept by the collector π ac2 U∞ n∞ . The capture efficiency is then given by η=

2π ψL n∞ 2ψL = 2 π ac2 U∞ n∞ ac U∞

(13.5)

13.2

CLASSICAL DEPOSITION MECHANISMS

473

Making use of Eq. (13.4), the interception capture efficiency becomes ηI =

 
1 ac3 3 1 2 a (a + a ) + + a ) − (a c p c p c ac2 2 2 ap + ac

(13.6)

For the case of ap /ac ≪ 1, the above expression reduces to 3 ηI = 2




ap ac

2

(13.7)

The interception capture efficiency ηI is normally used as a means to assess other modes of capture. For the case of flows with a high Reynolds number, the interception capture efficiency due to a spherical collector was given by Weber and Paddock (1983) as ηI =

3 2




ap ac

2 

1+

0.375Re 1 + 0.367Re0.56



for 0 < Re ≤ 150

(13.8)

where Re = U∞ ac ρ/µ. For the case of creeping flow (Re → 0) in a packed bed of spherical collectors, the interception capture efficiency of a single collector is modified as 3 ηI = 2




ap ac

2

(13.9)

Asph

where Asph is a dimensionless parameter that expresses the modification of the stream function due to the presence of other collector particles (Spielman, 1977; Neale and Masliyah, 1975). This type of approach has been quite successful in the modeling of capture by pure interception and under colloidal forces in a packed bed or a filter mat. For Happel’s cell model simulating flow through a packed bed of spheres, the dimensionless function Asph is provided by (Happel, 1958; Spielman and Fitzpatrick, 1973) 5/3

Asph =

2(1 − αc ) 1/3

2 − 3αc

5/3

+ 3αc

− 2αc2

(13.10)

where αc is the collector volume fraction. Note when αc → 0, Asph becomes unity. The collection efficiency for a single collector can be applied to evaluate the decrease in the particle number concentration through a packed bed or a filter mat. Assume a packed bed having nc spherical collectors per unit volume of the bed. Consider an element of thickness dx within the packed bed. A particle number balance on the differential thickness dx provides  dn [U∞ nAb ] − U∞ nAb + U∞ Ab dx − (nc Ab dx)(πac2 U∞ n)η = 0 dx 

(13.11)

474

DEPOSITION OF COLLOIDAL PARTICLES

Figure 13.3. Particle deposition in a packed bed of spherical collectors.

where U∞ is the fluid velocity away from the collector surface, Ab is the bed crosssectional area and n is the particle number concentration at distance x along the bed. The differential element is shown is Figure 13.3. The particle balance equation (13.11) reduces to dn = −π ac2 nc ηn (13.12) dx The number of collector grains per unit volume, nc , is given by nc =

3αc 4π ac3

(13.13)

Combining Eqs. (13.12) and (13.13) leads to 
3αc η dn n =− dx 4ac

(13.14)

For convenience, a filter coefficient is normally defined as λ=

3αc η 4ac

(13.15)

Making use of the filter coefficient definition, we obtain dn = −λn dx

(13.16)

13.2

CLASSICAL DEPOSITION MECHANISMS

475

For an incoming suspended particle number concentration of no (at x = 0), Eq. (13.16) can be integrated to give n = no exp(−λx)

(13.17)

Equation (13.17) indicates that the decay in the number concentration is exponential and that the filter coefficient λ can be considered as a characteristic penetration depth. The collision efficiency η can be determined for cases other than pure interception as will be shown in later sections. 13.2.3 Inertial Deposition When suspended non-interacting particles do not follow the flow streamlines due to their inertia, their capture by a collector is termed inertial deposition. For particles of less that 10 µm in liquids, inertial deposition is not important. However, inertial deposition in gases can be very significant (Friedlander, 2000). Figure 13.4 shows inertial deposition of a spherical particle onto a cylindrical collector. Newton’s second law can be used to write a force balance for a particle in a flow field in the absence of gravity and for ρp ≫ ρ. It is given as mp


 d 2x dx = 6π µa u− p dt 2 dt

(13.18)

where mp is the mass of the particles. In such a formulation it is assumed that the particle is much smaller than the collector and at sufficiently low concentration such that the flow field about the collector is not altered by the presence of the particles (Clift et al., 1978; Friedlander, 2000). The term on the left side of Eq. (13.18) is the particle acceleration; the term on the right side is the drag on the particle. Here Stokes drag is used because the particle Reynolds number is assumed to be small. The term u is the fluid velocity vector and x is the particle position vector. Procedures to solve Eq. (13.18) are given by Michael and Norey (1969), Griffin and Meisen (1973), and Masliyah and Duff (1975).

Figure 13.4. Inertial impaction of a spherical particle on a cylindrical fibre in the absence of gravity and for ρp ≫ ρ.

476

DEPOSITION OF COLLOIDAL PARTICLES

Non-dimensionalization of Eq. (13.18) leads to St

d 2X dX = u∗ − dτ 2 dτ

(13.19)

where τ=

U∞ t ac

(13.20)

x ac

(13.21)

u U∞

(13.22)

X= u∗ = and 2 St = 9




ap ac

2


ρp ρ




ρac U∞ µ



(13.23)

Here ρ and ρp are the fluid and particle densities, respectively. St is the Stokes number which is a measure of the particle inertia with respect to viscous forces. Variation of the collection efficiency η with the Stokes number is shown in Figure 13.5. The flow field used is that of a potential flow over a sphere given by   a 3  1 c 2 ψ = U∞ r 1 − sin2 θ 2 r The collection efficiency increases with increasing Stokes number where the particle inertia becomes important.

Figure 13.5. Deposition efficiency due to inertial capture by a spherical collector using potential flow velocity (Michael and Norey, 1969).

13.3

13.3

EULERIAN APPROACH

477

EULERIAN APPROACH

Capture of colloidal particles from a flowing suspension of particles by a stationary collector surface can occur due to the combination of Brownian motion, hydrodynamic interaction, and the presence of other forces, such as gravitational, London–van der Waals, and electric double layer forces. In this section, we present the methodology of analyzing particle capture or deposition employing the Eulerian approach. The general mathematical construct of the Eulerian approach is first presented, followed by consideration of specific geometries, and the different type of forces. The generalized picture of particle deposition on a stationary collector surface involves convective transport of particles with the fluid flow toward the collector, diffusive transport of the particles from regions of high to low concentrations, and migration due to other forces like gravity or colloidal forces. The generalized convection–diffusion equation which governs particle transfer due to a flow field characterized by a velocity vector u is given by ∂n +∇ ·j=Q ∂t where =

j = vn − D · ∇n +

n = D·F kB T

(13.24)

(13.25)

where j is the flux of the particles (particles per unit time and unit area), n is the local = particle number concentration, D is the diffusion coefficient tensor, F is the force experienced by the particle, v is the particle hydrodynamic velocity, i.e., the particle velocity that is induced by the fluid motion in the absence of all other forces such as Brownian or colloidal forces, T is the absolute temperature, kB is the Boltzmann constant and Q is a source term. It is noted that in Eq. (13.25), the hydrodynamic particle velocity, v, is used instead of the fluid velocity, u, as was discussed in Chapter 6, Section 6.3. In this chapter, we are discussing large particles, which cannot be treated as point masses. Consequently, when such finite sized particles are suspended in a fluid with a spatially non-uniform velocity field, different parts of the particle are subjected to fluid velocities of different intensities. As a result, the net hydrodynamic particle velocity determined at the particle center of mass becomes different from the fluid velocity. Particularly, such differences between the hydrodynamic particle velocity and the fluid velocity become pronounced for flows close to a stationary surface. It is for this reason that the fluid velocity, u, is replaced by the particle hydrodynamic velocity, v. It is this hydrodynamic particle velocity that needs to be adjusted by the diffusional fluxes and other external forces to arrive at the particle flux j. The reason for the introduction of the diffusion coefficient tensor rather than simply using a scalar quantity is that the particle diffusion coefficient is a function of the particle position relative to a surface. Once again, this artifact of incorporating particle hydrodynamic interactions was briefly discussed in Section 6.3. This was illustrated by Honig et al. (1971), who gave expressions for the diffusion coefficient variation

478

DEPOSITION OF COLLOIDAL PARTICLES

with gap width between two spherical particles. For the case of a spherical particle (1) diffusing near another spherical particle (2), one can write the diffusion coefficient tensor as 
= d 0 D = (D1 + D2 )∞ (13.26) 0 d⊥

where (D1 + D2 )∞ is the diffusion coefficient of the diffusing spherical particles in an infinite medium given by the Stokes–Einstein equation. The term d and d⊥ are correction factors due to the finite gap width between the two spheres. They are both functions of the ratio of the spheres radii and the separation gap width. For the case of equal-sized spheres, d⊥ = 1/G(h) as was given by Eq. (12.54). For the case of a spherical colloidal particle of radius a and a very large collector grain, which is treated as a flat surface, Eq. (13.26) becomes 
= d 0 D = D∞  (13.27) 0 d⊥ It should be noted, however, that d and d⊥ of Eq. (13.27) are different from those of Eq. (13.26) because d and d⊥ are functions of the sphere size ratio. Both d and d⊥ approach zero as the gap width h → 0. For small h d⊥ = 2h for a1 = a2 and d⊥ = h

for a2 → ∞

The correction function d is proportional to 1/| ln h| for h → 0. As the dimensionless gap width becomes large, d → 1 and d⊥ → 1 (Batchelor, 1976; Adler, 1981; Jeffrey and Onishi, 1984; van de Ven, 1989). Note that is some publications the definition of the terms d and d⊥ or their equivalent are interchanged. The velocity of the fluid u is usually evaluated from the solution of the Navier– Stokes equation in the absence of the dispersed particles. To this end, the analysis becomes limited to a very diluted dispersed phase. The required hydrodynamic velocity of the particle v is related to the fluid velocity u by making corrections due to the presence of the collector surface. For a deposition problem involving a specific collector geometry, the forces and the flow configuration are first identified. The particle flux j is then formulated in terms of the convection, diffusion, and migration contributions in Eq. (13.25). Finally, solution to Eq. (13.24) is obtained employing appropriate boundary conditions at the collector surface. In the following subsections, various situations will be presented to illustrate the use to the Eulerian approach. 13.3.1 Deposition Due to Brownian Diffusion Without External Forces: Spherical Collector Here we shall deal with the classical case of deposition onto a spherical collector due to Brownian diffusion alone in the absence of any other forces. The diffusing

13.3

EULERIAN APPROACH

479

particles are considered to be of submicron size (strictly speaking, point masses) with their hydrodynamic velocity identical to the fluid velocity (v = u). In this case, we ignore any hydrodynamic interactions. With the radius ap being very small (point particles), the colloidal forces are taken to be zero and the diffusion coefficient is constant. For steady-state conditions, with Q = F = 0 and for a constant diffusion coefficient, D∞ , combining Eqs. (13.24) and (13.25) leads to ∇·(nu−D∞ ∇n) = 0

(13.28)

Recognizing that ∇ · u = 0 for an incompressible fluid, the above equation becomes u · ∇n − D∞ ∇ 2 n = 0

(13.29)

Equation (13.29) in spherical coordinates becomes ur



 ∂ 1 ∂ ∂n ∂n uθ ∂n 1 ∂n + = D∞ 2 r2 + 2 sin θ ∂r r ∂θ r ∂r ∂r r sin θ ∂θ ∂θ

(13.30)

For Stokes flow about an isolated sphere (in the limit of a very small Reynolds number), the flow velocities in the radial and angular-directions are (Bird et al., 2002):   1 ac3 3 ac + ur = −U∞ cos θ 1 − 2 r 2 r3

(13.31)

  1 ac3 3 ac − uθ = U∞ sin θ 1 − 4 r 4 r3

(13.32)

 2  ∂ n 2 ∂n ∂n uθ ∂n + = D∞ + ur ∂r r ∂θ ∂r 2 r ∂r

(13.33)

and

For large Peclet numbers, the variation in the concentration is much higher in the r-direction that in the θ-direction and one can rewrite Eq. (13.30) as

The flow and the coordinate system is shown in Figure 13.6. Assuming a perfect sink at the spherical collector surface, the boundary conditions become symmetry at θ = 0, π n=0

at r = ac

for all r

for all θ

(13.34) (13.35)

and n → n∞

as r → ∞ for all θ

(13.36)

Here n∞ is the bulk value of n at a large distance from the spherical collector. A solution to Eq. (13.33) is given by Masliyah and Epstein (1973) and other solutions

480

DEPOSITION OF COLLOIDAL PARTICLES

Figure 13.6. Diffusion boundary layer around a spherical collector.

are cited by Clift et al. (1978). However, simplification in the solution can be made by assuming that the diffusion process is limited to a region very close to the surface and hence the velocity profile needs to be true only close to the spherical collector. Consequently, the velocity components are approximated by 3 ur = − 2




y ac

2

3 2




y ac



U∞ cos θ

(13.37)

and uθ =

U∞ sin θ

(13.38)

where r y =1+ ac ac

(13.39)

The above velocity expressions are valid close to the collector surface. Levich (1962) obtained a similar solutions to Eqs. (13.33) to (13.39). At the collector surface, the flux component normal to the collector surface, j⊥ , is given by  ∂n  j⊥ = −jr = D ∂r surface

(13.40)

in m−2 s−1 , and the total rate of particles diffusing to the surface is given by 

S

j⊥ dS =

2π ac2



0

π

j⊥ sin θ dθ

(13.41)

where j⊥ is the flux normal to the collector surface and S is the spherical collector surface area.

13.3

EULERIAN APPROACH

481

The local Sherwood number characterizing the mass transfer is defined as Sh(θ) =

a c j⊥ D ∞ n∞

(13.42)

where n∞ is the particle number concentration at a large distance from the surface of the spherical collector. The average Sherwood number is defined as  1 Sh(θ) dS (13.43) Sh = S S The solution of Eq. (13.33) with the velocities defined by Eqs. (13.37) through (13.39) subject to the boundary conditions of Eqs. (13.34) through (13.36) provides the dimensionless mass transfer number (Clift et al., 1978): Sh = 0.624(Re Sc)1/3

for ReSc ≫ 1

(13.44)

where the Schmidt number (ratio of momentum transfer to molecular diffusion) is given by µ Sc = (13.45) ρD∞ and the Reynolds number (ratio of inertial to viscous forces) is given by Re =

ac ρU∞ µ

(13.46)

Equation (13.44) is usually referred to as the Lighthill–Levich equation or simply the Levich equation.2 The product of Re and Sc is normally called the Peclet number and it is given by ac U∞ P e = ReSc = (13.47) D∞

Acrivos and Goddard (1965) gave a first-order correction to Eq. (13.44) as Sh = 0.624(P e)1/3 + 0.46 P e ≫ 1

(13.48)

For the case of small Peclet numbers, solution of the complete convection-diffusion equation for the case of creeping flow is given by Acrivos and Taylor (1962) as Sh = 1 +

2

Pe 2

Pe ≪ 1

(13.49)

The original derivation of Levich for P e → ∞ was given as Sh = (7.98/4π )P e1/3 , where the coefficient becomes 0.635. This value of the coefficient is slightly higher than in the more recent analysis, Clift et al. (1978).

482

DEPOSITION OF COLLOIDAL PARTICLES

Equations (13.44), (13.48), and (13.49) were derived using creeping flow velocities (Re → 0) about the spherical collector and the equations are identical to those given for classical mass transfer from spheres. Equation (13.49) states that for pure diffusion, the Sherwood number is unity. In the definition of both Peclet and Sherwood numbers, the radius of the diffusing particle is used as the characteristic length. 13.3.2 Deposition due to Brownian Diffusion with External Forces: Stagnation Flow Convenient techniques that are often used in experimental and theoretical studies for deposition of colloidal particles are those of rotating disks (Marshall and Kitchener, 1966; Hull and Ketchner, 1969; Prieve and Lin, 1980) and of stagnation flows due to impinging jets (Dabros and van de Ven, 1983; Boluk and van de Ven, 1989; Adamczyk et al., 1989). Other geometries are discussed by Adamczyk (1989a,b) and Elimelech et al. (1995). In the previous section, diffusion due to Brownian motion was considered with the diffusing particles taken as “point” particles. In this section we shall discuss in detail the Eulerian approach to the study of finite-sized particle deposition in a stagnation flow due to an impinging jet. We shall take into account the relationship between the fluid and particle velocities, a variable diffusion coefficient, and interparticle forces. Consider a fluid containing colloidal particles impinging on a flat plate as shown in Figure 13.7. Due to the axisymmetry of the flow, the velocity components in the angular direction are zero and the velocity field is given by the radial and the normal components, namely ur and uz , respectively. As the Schmidt number of the colloidal system exceeds unity by one or two orders of magnitude, the diffusion boundary layer

Figure 13.7. Impinging jet geometry. (a) Overall flow geometry. (b) Velocity profile at the stagnation region shown by the dashed box in (a). O is the stagnation point where the velocity is zero. (c) Position of a colloidal particle with respect to the impinging surface.

13.3

EULERIAN APPROACH

483

is much thinner than the hydrodynamic boundary layer. It is for this reason that the flow velocities need to be characterized close to the impingement region only. For flow near the stagnation region, the velocities of the fluid are given by ur = αzr

(13.50)

uz = −αz2

(13.51)

and

The parameter α characterizes the intensity of the flow (Dabros and van de Ven, 1983). Here it is assumed that the presence of the particles does not alter the flow field. The movement of a particle entrained in the stagnation region of the flow is decomposed into normal (z-direction) and tangential (r-direction) components. The colloidal particle Reynolds number is assumed to be sufficiently small such that Stokes flow holds. Equation (13.25) can be written as

and




∂n n ∂n + Drz Drr Fr + Drz Fz + jr = vr n − Drr ∂r ∂z kB T


∂n ∂n + Dzz jz = vz n − Dzr ∂r ∂z



+

n [Dzr Fr + Dzz Fz ] kB T

(13.52)

(13.53)

Note that in the above expressions, the product of the diffusion coefficient and the = force components represents a product of the diffusion coefficient tensor, D, with the = force vector, F. The dot product of a tensor T with a vector A is given by =

T · A = (Txx Ax + Txy Ay + Txz Az )ix + (Tyx Ax + Tyy Ay + Tyz Az )iy + (Tzx Ax + Tzy Ay + Tzz Az )iz . Recognizing from Eq. (13.27) that Dzr = Drz = 0, we can write jr = vr n − Drr

∂n n + Drr Fr ∂r kB T

(13.54)

jz = vz n − Dzz

n ∂n + Dzz Fz ∂z kB T

(13.55)

and

where jr and jz are the fluxes in r- and z-directions, respectively. Following Spielman and Fitzpatrick (1973), Prieve and Lin (1980), Dabros and van de Ven (1983), and van de Ven (1989), the relationships between the fluid and

484

DEPOSITION OF COLLOIDAL PARTICLES

TABLE 13.1. Asymptotic Expressions for the Universal Hydrodynamic Functions for a Spherical Particle Near a Flat Solid Surface (Spielman and Fitzpatrick, 1973). Function

h→0

H →∞

f1 (h)

h

1−

f2 (h)

3.23



f3 (h)

0.7431 0.63736 − 0.2 ln h

9 8(h + 1)

−1 9 8(h + 1) 5 1− 16(h + 1)3 1−

particle velocities and the local particle diffusion coefficients are given by (13.56)

vr = ur f3 (h)

(13.57)

vz = uz f1 (h)f2 (h) Drr = D∞ d = D∞ f4 (h)

(13.58)

Dzz = D∞ d⊥ = D∞ f1 (h)

(13.59)

and where f1 to f4 are the universal hydrodynamic (correction) functions relating the deviation from Stokes flow and the Stokes–Einstein relationship due to the presence of the collector wall. f1 is given by Brenner (1961), f2 is given by Goren (1970) and Goren and O’Neill (1971), and f3 and f4 are given by Goldman et al. (1967a,b). Asymptotic behaviors of f are listed in Table 13.1. Curve-fit expressions for f are given in Table 13.2. Figure 13.8 shows the variation of the functions with the dimensionless gap h, where z =h+1 (13.60) ap It should be noted that the correction factors f1 to f4 and d and d⊥ are for the geometry of a sphere and a solid plane as applicable to the problem at hand. Making use of Eqs. (13.56) to (13.59), the flux Eqs. (13.54) and (13.55) become jr = ur f3 n − D∞ f4

∂n n + D∞ f4 Fr ∂r kB T

(13.61)

TABLE 13.2. Curve Fit for the Universal Functions fi . i

a

b

c

d

e

1 2 3 4

0.9267 0.5695 0.15 1.26

−0.3990 1.355 −0.375 −2.676

0.1487 1.36 3.906 0.3581

−0.601 0.875 −0.625 1.999

1.202 0.525 3.105 0.232

fi = 1.0 + bi exp(−ci h) + di exp(−ei hai )

13.3

EULERIAN APPROACH

485

Figure 13.8. Universal hydrodynamic correction functions for diffusion and particle motion near a plane for the case of a sphere.

and jz = uz f1 f2 n − D∞ f1

∂n nD∞ + f 1 Fz ∂z kB T

(13.62)

where D∞ is given by the Stokes–Einstein equation. The z-direction component of the force Fz is composed of gravitational force, London–van der Waals force, and electrostatic (electric double layer) force. Fz is given by Fz = Fgz + FA + FR

(13.63)

Here it is assumed that the colloidal forces act normal to the surface of the collector. The r-direction component of the force is due to gravity alone: Fr = Fgr

(13.64)

For a horizontal deposition plate, Fr is zero. For steady-state conditions and a zero source term, Eq. (13.24) in cylindrical coordinates becomes ∂jz 1 ∂ (rjr ) + =0 (13.65) r ∂r ∂z From Eqs. (13.61) and (13.62), the flux conservation Eq. (13.65) becomes 
 n ∂n 1 ∂ + D∞ f4 Fr r ur f3 n − D∞ f4 r ∂r ∂r kB T   ∂ n ∂n + D∞ f1 Fz = 0 + uz f1 f2 n − D∞ f1 ∂z ∂z kB T

(13.66)

486

DEPOSITION OF COLLOIDAL PARTICLES

Equation (13.66) is the convection-diffusion equation for the transport process in the presence of external forces. This equation was derived for a stagnation flow configuration and is also applicable to other axisymmetric deposition configuration such as a rotating disc. The elements that make up the forces Fr and Fz are discussed below. The gravitational force is given by Fgr =

4π 3 a ρg sin θ 3 p

(13.67)

and

4π 3 a ρg cos θ (13.68) 3 p where ρ is the density difference between the particle and the fluid. The inclination angle θ is between the z-axis and the vertical direction. The attractive interaction force, FA , is typically represented by an appropriate expression for the van der Waals forces (see Chapter 11). A convenient expression for the retarded London–van der Waals dispersion force between a spherical particle and a planar collector surface is given by (Suzuki et al., 1989) Fgz =

− −

AH λ(λ + 22.232h) − FA = − 6ap h2 (λ + 11.116h)2

(13.69)

where AH is the effective Hamaker constant for interaction between the particle and − the collector in the suspending medium (= A123 ) and λ = λ/ap is a dimensionless parameter accounting for electromagnetic retardation, with λ being the London wavelength. Usually, the value of the London wavelength is taken as 10−7 m (100 nm). The repulsive interaction force, FR , between the particle and the collector predominantly arises due to the electrical double layer interactions. For spherical particles and planar collectors, the electrical double layer force expression is given by (Hogg et al., 1966; Usui, 1973)   exp(−κap h) (ζc − ζp )2 exp(−2κap h)

FR = 4π ǫκap ζp ζc ∓ 1 ± exp(−κap h) 2ζc ζp 1 − exp(−2κap h) (13.70)

The upper and lower signs in Eq. (13.70) correspond to constant surface potential and constant surface charge, respectively. The following dimensionless groups are defined as: Gravitational parameter, Gr =

gravitational force Brownian motion force

Gr = where D∞ = kB T /(6π µap ).

2 ρgap3 9 µD∞

(13.71)

13.3

Double layer parameter, Dl =

EULERIAN APPROACH

487

electrostatic force Brownian motion force

Dl =

4π ǫap ζp ζc kB T

(13.72)

Double layer asymmetry parameter, Da, referring to the asymmetry of the double layers for different surface potentials on the interacting surfaces, defined as Da =

(ζc − ζp )2 2ζc ζp

(13.73)

For ζc = ζp , Da becomes zero. Dimensionless inverse Debye length, κap , given by κap =




2000z2 e2 NA M ǫkB T

1/2

ap

(13.74)

where NA is the Avogadro number, and M is the molar concentration of the symmetric (z : z) electrolyte. Normalizing the forces by (kB T /ap ), neglecting radial derivatives, and setting θ = 0, Eq. (13.66) becomes   d 1 dn P ef1 f2 n(1 + h)2 + f1 − nf1 Fz∗ − P ef3 (1 + h)n = 0 dh 2 dh

(13.75)

where Fz∗

− −  exp(−κap h) λ(λ + 22.232h) − = −Ad 2 + (Dl)(κap ) h (λ + 11.116h)2 1 ± exp(−κap h)  exp(−2κap h) ∓ Da + Gr {1 − exp(−2κap h)}

(13.76)

In Eqs. (13.75) and (13.76) Pe =

2ap3 α D∞

(13.77)

and Ad =

AH 6kB T

(13.78)

Ad is a measure of the relative strengths of London–van der Waals force to the Brownian motion force. It is usually referred to as the adhesion number or parameter.

488

DEPOSITION OF COLLOIDAL PARTICLES −

The parameters that influence particle deposition on smooth surfaces are κa, λ, Ad, Dl, Da, Gr, and P e. Although the above analysis was derived for stagnation flow, it is valid for any flow configuration having velocities of the form given by Eqs. (13.50) and (13.51). Therefore by using an appropriate definition for the Peclet number, P e, the analysis for various flow geometries becomes similar. For the case of an impinging jet, the flow strength α is given by Re ν − α = α(Re) 3 Rj et −

where α(Re) = βRe1/2 , and β is of order unity (Dabros and van de Ven, 1987). Rj et is the jet tube radius and ν = µ/ρ is the kinematic viscosity. The Reynolds number Re is given by U∞ Rj et (13.79) Re = ν where U∞ is the average exit velocity of the jet. Various definitions of Peclet number for other geometries are given by van de Ven (1989). The boundary conditions of Eq. (13.75) are as h → ∞

(13.80)

at h = δm /ap

(13.81)

n → n∞ and n→0

where δm is generally identified with the distance of primary energy minimum. The solution of Eq. (13.75) subject to Eqs. (13.80) and (13.81) can be expected to yield an upper limit to the flux towards the collector for a given energy of interaction between the particle and surface and for given hydrodynamic conditions (Dabros and van de Ven 1983, 1987). The boundary condition [Eq. (13.81)] assumes that all particles arriving close to the collector surface are irreversibly captured in an infinite-depth energy sink and disappear from the system (Adamczyk, 1989a). It is often referred to as the perfect sink boundary condition. In most studies, δm /ap is taken as a fixed small value, say 10−3 (Prieve and Lin, 1980), since setting δm = 0 in conjunction with any of the Hamaker type expressions for the van der Waals interactions will lead to a divergence of the interaction forces to −∞ at contact. The local Sherwood number is defined as Shl = −jzw

ap ap = j⊥ D∞ n∞ D ∞ n∞

where jzw is the flux in the z-direction at the surface of the collector (i.e., at z = a + δm ). The average Sherwood number is given by 1 Sh = S



S

Shl dS

(13.82)

13.3

EULERIAN APPROACH

489

where Sh is the average Sherwood number and S is the deposition surface area. jzw is evaluated using Eq. (13.62) once the distribution on n is known. Numerical techniques are normally employed in solving Eq. (13.75), thus providing the variation of n with h. Transport equations similar to Eq. (13.75) are given by Prieve and Ruckenstein (1974) for the case of a spherical collector; Prieve and Lin (1980), Rajagopalan and Kim (1981), and Clint et al. (1973) for the case of a rotating disc; Dabros and van de Ven (1983, 1987) and van de Ven (1989) for the case of an impinging jet; Adamczyk and van de Ven (1981a) for the case of parallel plates channel; and Adamczyk and van de Ven (1981b) for the case of cylindrical collectors. General discussions on the kinetics of particle accumulation on collectors is given by Adamczyk et al. (1984), Adamczyk and van de Ven (1984), and Adamczyk et al. (1992). Solution of the transport equations such as Eq. (13.75) provides values for the Sherwood number Sh and a knowledge of how the various parameters, e.g., P e, Dl, Da, and Ad affect mass transport. In this section, detailed analysis was given for the impinging jet geometry. Mass transfer analysis, i.e., particle deposition for other geometries, is very similar. The effect of the various parameters on the dimensionless mass transfer Sh will be given below. Table 13.3 gives the definitions of the Reynolds, Peclet, and Sherwood numbers for different geometries employed in the literature. van de Ven (1989) provides tabulation of the velocity components near the surface of the various geometries. For the case of Brownian particles with negligible London–van der Waals and electric double-layer forces, the stagnation flow Sherwood number is given by the Levich-type equation as Sh = 0.616P e1/3 for P e ≫ 1 and Dl = Ad = Gr = 0, where P e and Sh are defined by Eqs. (13.77) and (13.82), respectively. Sherwood number variation is similar to that obtained for mass transfer of point particles towards a spherical collector as given by Eq. (13.44).

TABLE 13.3. Definition of Reynolds, Peclet, and Sherwood Numbers for Various Deposition Geometries. Collector Geometry

Reynolds Number

Peclet Number

Spherical collector

ac ρU∞ µ ac ρU∞ µ Rj et ρU∞ µ  ω 3/2 ap3 υ

ac U∞ D∞ ac U∞ D∞ 2ap3 α

Cylindrical collector Impinging jet Rotating disc

D∞ 1.02ω3/2 ap3 υ 1/2 D∞

1 S 1 S 1 S 1 S

Sherwood Number  
ac j⊥ D∞ n∞  S
ac j⊥ D∞ n∞  S
ap j⊥ D∞ n∞ S  
ap j⊥ D∞ n∞ S

dS dS dS dS

490

DEPOSITION OF COLLOIDAL PARTICLES

Figure 13.9. Calculated values of the Sherwood number for stagnation flow as a function of the double layer parameter Dl (Dabros and van de Ven, 1983).

Dabros and van de Ven (1983) gave the variation of Sherwood number, Sh, with the double-layer parameter, Dl, for various Reynolds numbers, Re. A negative Dl number means that the particle and the collector have opposite signs for their surface (zeta) potentials. It is clear from Figure 13.9 that a critical Dl value exists. Beyond the critical Dl value the deposition rate is suddenly reduced by several orders of magnitude. By increasing Dl, the energy barrier increases and the height of the energy barrier becomes the rate-determining factor. This type of sudden drop in Sh with Dl is characteristic of all collectors and is also observed during coagulation of colloidal particles (van de Ven, 1989). Figure 13.9 also shows that changing the Reynolds number from 10 to 30 has the effect of increasing Sh when deposition occurs, but has an insignificant effect when the deposition rate is small. Other examples for the effect of Dl are given by Adamczyk (1989a,b). Figure 13.10 shows the dependence of the Sherwood number, Sh, on Reynolds number, Re, for various values of the double layer parameter, Dl, for λ¯ = 0.4, Ad = 0.8, κap = 58.2, Gr = 0.002, and Da = 0 for stagnation flow collector. When Dl = 0, where no electrostatic repulsion exists, the flux of the particles, as represented by the Sherwood number, to the collector is high and depends on the Reynolds number, unlike the case when a high energy barrier exists for Dl > 0. When Dl = 110, the Sherwood number becomes quite small for all Reynolds numbers tested (Dabros and van de Ven, 1983). 13.3.3 Deposition due to Brownian Diffusion with External Forces: Spherical Collectors Deposition analysis due to Brownian diffusion with external forces for spherical collectors was made by Prieve and Ruckenstein (1974). In their study, Prieve and

13.3

EULERIAN APPROACH

491

Figure 13.10. Theoretical values of the Sherwood number for stagnation flow as a function of the jet Reynolds number for Ad = 0.8, λ¯ = 0.4, Gr = 0.002, and κap = 58.2 (Dabros and van de Ven, 1983).

Ruckenstein considered a packed bed made up of spherical collectors. They used the Spielman and Fitzpatrick (1973) approximation of Happel’s cell model to describe the flow in a packed bed of spherical particles (see Figure 13.6 for coordinate system). The exact Happel’s cell model that provides the stream function of the flow field around a spherical collector embedded in the packed bed is given by 
4 
2
 1 r r r A ψ = U∞ +D ac2 sin2 θ (13.83) +B +C 2 (r/ac ) ac ac ac where A = 1/w

(13.84-a)

B = −(3 + 2αc5/3 )/w

(13.84-b)

C = (2 + 2αc5/3 )/w

(13.84-c)

D = −αc5/3 /w

(13.84-d)

w = (2 − 3αc1/3 + 3αc5/3 − 2αc2 )

(13.84-e)

and

Here αc is the volume fraction of solids in the packed bed, U∞ is the approach fluid velocity, and ac is the collector radius.

492

DEPOSITION OF COLLOIDAL PARTICLES

The fluid velocity components are given by ur = −

1 ∂ψ r 2 sin θ ∂θ

(13.85-a)

and uθ =

1 ∂ψ r sin θ ∂r

(13.85-b)

To simplify the analysis for flow in a packed bed of spheres, Spielman and Fitzpatrick (1973) introduced a flow model parameter, Asph , for the case of spherical collectors in a packed bed defined as 5/3

Asph =

2(1 − αc ) 1/3

2 − 3αc

5/3

+ 3αc

− 2αc2

(13.86)

together with an approximate expression for the stream function that is valid near the collector. The approximate expression for the stream function is given by ψ=

3 Asph U∞ [r − ac ]2 sin2 θ 4

(13.87)

The factor Asph accounts for the influence of neighboring spheres on the fluid flow field. The expression for Asph would vary according to the model used to describe the flow over a bed of spherical collectors (Tien, 1989). Using an approach similar to that presented in the previous section for stagnation flow, Prieve and Ruckenstein (1974) solved the flux equations equivalent to that given by Eq. (13.75) in the presence of gravity and van der Waals forces. Electrostatic forces were not included in their analysis. As their study included packed bed of spheres, the Peclet number can be conveniently defined as3 Pe =

ac U∞ Asph D∞

(13.88)

to allow for the effects of neighboring spheres. The Sherwood number is given by Eq. (13.42). Prieve and Ruckenstein (1974) used in their analysis fluid velocities given by (see Figure 13.6 for coordinate system)   1 3 3 ur = −Asph U∞ 1 − (ac /r) + (ac /r) cos θ (13.89-a) 2 2 and   1 3 uθ = Asph U∞ 1 − (ac /r) − (ac /r)3 sin θ 4 4

(13.89-b)

3 In the original paper by Prieve and Ruckenstein, (1974) Sherwood and Peclet numbers were based on the collector diameter, ac .

13.3

EULERIAN APPROACH

493

Figure 13.11. Sherwood numbers computed for the convection–diffusion of particles of finite size to the surface of a spherical collector by neglecting interaction forces. The dashed line is for the Levich equation which is valid when a diffusion boundary layer exists and the particles are infinitesimal (Prieve and Ruckenstein, 1974).

For a spherical collector, in the absence of external forces, the effect of a finite particle size on the mass transfer (deposition) is shown in Figure 13.11. Here the Sherwood number is given for the transport of particles of finite size to the surface of a spherical collector. The Levich-type equation, Eq. (13.44), which is valid for ap /ac → 0 and when a diffusion boundary layer exists (P e ≫ 1), is plotted for comparison. Deviation from the Levich equation occurs when the diffusion boundary layer thickness is not small compared to the collector radius. This occurs at low values of P e. Also deviation from the Levich equation becomes apparent when the particle size becomes large as compared with the diffusion boundary layer thickness. This situation occurs at high values of P e. For the case of a spherical collector under no-flow conditions, i.e., P e = 0, the effect of London–van der Waals forces on the Sherwood number is shown in Figure 13.12 for finite size particles. It is clear from Figure 13.12 that when ac /ap becomes large, Sh → 1 irrespective of the values of AH /kB T , which are a measure of the attractive dispersion force. However, for smaller values of ac /ap , the dispersion force has a large influence on the Sherwood number which is a measure of the deposition (Prieve and Ruckenstein, 1974). The limit of Sh → 1 as P e → 0 is that of pure diffusion of infinitesimal particles. For the case of a spherical collector, the effect of the London–van der Waals forces on the deposition in the absence of other forces was studied by Prieve and Ruckenstein (1974). They evaluated the Sherwood number as a function of the Peclet number for different values of ap /ac . Figure 13.13 shows the case of ap /ac = 10−4 . The Levich Eq. (13.44) is shown for comparison. At low values

494

DEPOSITION OF COLLOIDAL PARTICLES

Figure 13.12. Sherwood numbers computed for the transport of finite sized particles through a stagnant fluid to a spherical collector under the action of diffusion and van der Waals forces (Prieve and Ruckenstein, 1974).

of Peclet number, Sh becomes insensitive to AH /kB T . This is because the deposition is diffusion-controlled. At higher values of P e, the Sherwood number becomes more dependent on AH /kB T . This is due to the fact that the diffusion boundary layer becomes very thin and offers little resistance to mass transfer, consequently the

Figure 13.13. Sherwood numbers computed for the transport of finite particles to a spherical collector under the combined influence of diffusion and van der Waals forces (Prieve and Ruckenstein, 1974).

13.3

EULERIAN APPROACH

495

dispersion force plays a more important role in the mass transfer process. The reasons for the deviation from the Levich equation at very small and large P e are similar to those advanced for Figure 13.11. Plots for the case of a rotating disc are given by Prieve and Lin (1980). For large values of P e and (AH /kB T )/P e, the deposition Sherwood number in the absence of electrostatic and gravity forces is given by


AH Sh = 0.55 kB T

1/3

P e2/3

where the terms Sh and P e are defined in Table 13.3. Prieve and Ruckenstein (1974) gave an approximate plot indicating the regions where dispersion or convection-diffusion is rate controlling for the case of a spherical collector. Figure 13.14 identifies the regions in which either London–van der Waals or diffusion may be neglected when calculating the deposition rate. When the diffusion boundary layer is thin, the dispersion forces control the deposition rate. On the other hand, when the diffusion boundary layer is thick, the convection–diffusion process controls the deposition rate. For a spherical collector, the effect of gravitational forces upon the rate of deposition is shown in Figure 13.15 in the absence of an electric double-layer force. The gravitational field is in the main flow direction. The effect of gravitational force is substantial at lower values of P e. The gravitational force is characterized by the gravitational parameter which is defined as the ratio between the gravitational force to

Figure 13.14. Regions of capture mechanism and controlling rate for the case of a spherical collector for ac /ap > 103 in the absence of electrostatic double layer forces (Prieve and Ruckenstein, 1974).

496

DEPOSITION OF COLLOIDAL PARTICLES

Figure 13.15. Effect of gravitational forces upon the rate of deposition for a given radius ration for the case of spherical collectors in the absence of electrostatic force (Prieve and Ruckenstein, 1974).

that of thermal Brownian force, where Gr =

4π ap4 ρg 4π ap3 ρg = 3 kB T /ap 3kB T

Here, ρ is the density difference between the particle and the fluid. In the limit of Gr ≫ 1, the rate of particle deposition becomes π ac2 vt n∞ , particles/unit time where vt is the Stokes terminal velocity of the particle given by vt =

2gρap2 9µ

(13.90)

The deposition Sherwood number is then given by Sh =




π ac2 vt n∞ 4π ac2



ac D ∞ n∞

(13.91)



Gr

(13.92)

leading to 1 Sh = 4




ac ap

The flat regions of the curves of Figure 13.15 are given by Eq. (13.92). The question of whether adding the individual process rates produces the total effective mass transfer rate was addressed by Prieve and Ruckenstein (1974). They showed that there is good agreement between the exact solution of the transport equation and that obtained from the summation of the individual contributions to

13.4

LAGRANGIAN APPROACH

497

Figure 13.16. Adequacy of the additivity rule for ac /ap = 104 and AH /kB T = 104 . The curve represents results obtained by summing individual contributions from gravitational forces, Brownian motion, and van der Waals force. Filled circles were obtained from simultaneously considering all effects (Prieve and Ruckenstein, 1974).

the deposition rate from each mechanism. For the case of a spherical collector with ac /ap = 104 , and AH /kB T = 104 , individual contributions due to gravity, Brownian motion (diffusion) and dispersion forces were summed and compared with the exact solution that considers these effects simultaneously. The comparison between the two approaches is shown in Figure 13.16. The solid circles are given by the exact solution and the solid line is from the addition of individual contributions. The agreement between the two approaches is fairly good. However, there is no theoretical basis for concluding that the concept of additivity should hold under other physical parameters. It should be recognized that the comparison is made for AH /kB T = 104 which gives a rather large value for AH . 13.4

LAGRANGIAN APPROACH

In the previous section, we dealt with the deposition of particles having a finite Peclet number by solving the generalized convection-diffusion equation. Here we shall deal with particle capture (or deposition) in the absence of Brownian motion and hence diffusion, i.e., P e → ∞. The special case of particle capture by a spherical collector will be treated in detail. 13.4.1 Particle Collisions on a Spherical Collector: With the Presence of External Forces The case of particle capture by a spherical collector is of great practical importance in a variety of industrial applications, such as, in flotation columns where air bubbles

498

DEPOSITION OF COLLOIDAL PARTICLES

are used as collectors of small mineral particles, in filtration using a packed bed, in analysis of colloid transport in groundwater, and in chromatographic separations. In the Lagrangian approach, the flow field about the collector is assumed to be undisturbed by the presence of the colloidal particles. The fluid flow near the collector is deduced from the general expression of the flow and it is decomposed into a normal component and a tangential component with respect to the collector surface. A force balance on the colloidal particle is made in the directions normal and tangential to the collector surface. From the force balance equations, the angular variation of the gap between the colloidal particle and the surface is obtained and the trajectory equations provide the basis for the evaluation of deposition efficiency. When the spherical collector is small, it is possible to neglect the fluid inertia and the flow field is given by the Stokes solution. The undisturbed axisymmetric flow field close to a spherical collector is approximated by ψ=

3 Asph U∞ (r − ac )2 sin2 θ 4

(13.93)

where ψ is the stream function, U∞ is the undisturbed fluid velocity, ac is the collector radius, r is the radial coordinate and Asph is a dimensionless parameter characterizing the flow model used to account for the presence of other collectors. For an isolated spherical collector, Asph = 1. For a packed bed of spherical particles, Asph is given using Happel’s (1958) model by Spielman and Fitzpatrick (1973) as 5/3

Asph =

2(1 − αc ) 1/3

2 − 3αc

5/3

+ 3αc

− 2αc2

where αc is the collector grain volume fraction in the packed bed. Figure 13.17 shows the flow geometry together with the collector and the colloidal particle. The fluid velocity components are related to the stream function as 1 ∂ψ r 2 sin θ ∂θ

(13.94)

1 ∂ψ r sin θ ∂r

(13.95)

ur = − and uθ =

Following the approach of Spielman and FitzPatrick (1973), it is possible to define a system of local cylindrical coordinates ̟ and z whose origin is on the collector surface with r = ac and θ. Here θ is the angle corresponding to the center position of the colloidal particle to be captured. With this coordinate system, the origin of the coordinates changes position as the entrained particle moves around the collector. The fluid flow near the collector in the new cylindrical coordinates is given by u = ust + ush

(13.96)

13.4

LAGRANGIAN APPROACH

499

Figure 13.17. Flow geometry of a colloidal particle in the proximity of a larger spherical collector.

where ust =

3Asph U∞ cos θ [̟ zi̟ − z2 iz ] 2ac2

(13.97)

3Asph U∞ sin θ ziy 2ac

(13.98)

and ush =

where iy , i̟ , iz are unit vectors in the y, ̟, and z-directions, respectively. The velocity vector ust represents an axisymmetric stagnation flow at ̟ = 0, whereas ush represents a shear flow parallel to the collector surface. Locally, one can think of the entrained colloidal particle as being subjected to a stagnation flow and a shear flow as described by Eqs. (13.97) and (13.98). Both ust and ush represent the undisturbed fluid flow velocities about the collector sphere in the absence of the colloidal particle. This decomposition of the flow field is depicted in Figure 13.18. In part (a) of Figure 13.18, the overall geometrical representation of the variation in the deposition behavior from a completely stagnation point flow regime (when θ = 0 to a completely parallel flow regime (for θ = 90◦ ) is depicted. At intermediate angular positions, the deposition is governed by both stagnation and parallel flow. The stagnation flow and the shear flow components are shown in Figure 13.18(b) and (c), respectively. The entrained particle velocity is decomposed separately into flow fields corresponding to its normal and tangential velocities. This is allowed as the flow equation is linear and the method of superposition can be used (Happel and Brenner, 1965).

500

DEPOSITION OF COLLOIDAL PARTICLES

Figure 13.18. Flow geometry of a particle in the proximity of a larger spherical collector. (a) Overall geometry. The radial and angular velocity components of the fluid, ur and uθ , respectively, are shown for the spherical coordinate system with the collector center as the origin. This velocity field can be represented as linear superposition of two types of flow, namely, stagnation flow shown in part (b) and shear flow shown in (c).

The particle normal velocity ds/dt under the influence of a total normal force Fn is given by Fn f1 ds = dt 6π µap

(13.99)

where Fn is the total force acting on the entrained particle normal to the collector surface. Here f1 is the hydrodynamic universal correction factor given by Brenner (1961). It relates to the additional retardation in the translation velocity of the colloidal particle in a stagnant fluid towards a flat surface under the influence of a force Fn . Here s is the dimensional separation distance given by s = z − ap

(13.100)

The total force acting on the entrained particle normal to the collector surface is the sum of the London–van der Waals, electric double layer, gravitational, and

13.4

LAGRANGIAN APPROACH

501

hydrodynamic forces. The normal force Fn is given by Fn = FA + FR + Fgz + FHyd

(13.101)

The London–van der Waals force between a spherical particle and a plane is given by Eq. (13.69), the corresponding electrical double-layer force is given by Eq. (13.70), and the gravitational force is given by Fgz = −

4π 3 a ρg cos θ 3 p

(13.102)

The direction of the gravity force is assumed to coincide with the main direction of the flow. The hydrodynamic force is due to the fluid stagnation flow and it is given by FHyd

  3(s + ap )2 Asph U∞ cos θ (6π µap )f2 = − 2ac2

(13.103)

where the term in the first square brackets of Eq. (13.103) represents the z-directed fluid velocity as given by Eq. (13.97). To simplify the analysis we shall use the non-retarded expression for the London– van der Waals potential and assume the collector and the particle to have the same value for the zeta potential. This will allow the dropping of the second term in Eq. (13.70). Letting h = s/ap , Eq. (13.99) when combined with the constitutive expressions for the normal force components, yields dh dt



6π µap2 f1



2AH 3ap (h + 2)2 h2   exp(−κap h) 4π 3 + 4π ǫκap ζp ζc a ρg cos θ − 1 ± exp(−κap h) 3 p     3(h + 1)2 Asph U∞ cos θ 3 (13.104) − 6π µap f2 2ac2

=−

The above equation relates the variation of the dimensionless separation gap, h, with time to the various flow conditions and physical properties of the system. The separation gap is normalized by the spherical particle radius, ap . The motion of the entrained colloidal particle parallel to the collector (y- direction) is composed of two parts. The first part is the free rotation and translation due to the fluid shear velocity ush . The particle velocity due to the fluid shear velocity is given by vpy1

 3Asph U∞ (s + ap ) sin θ = f3 2ac 

(13.105)

The quantity in the square brackets of Eq. (13.105) is the shear velocity given by Eq. (13.98). The second part of the y-direction velocity of the particle corresponds

502

DEPOSITION OF COLLOIDAL PARTICLES

to the free rotation and translation of the particle under the external force of gravity and it is given by    4π ap3 ρg sin θ f4 (13.106) vpy2 ≃ 6π µap 3 The functions f3 and f4 are given by Goldman et al. (1967a,b). The resultant particle velocity in the y-direction is given by combining Eqs. (13.105) and (13.106) leading to ac

2ap2 ρg sin θ 3ap Asph U∞ (h + 1) sin θ dθ = f4 f3 + dt 2ac 9µ

(13.107)

The trajectory equation for the entrained particle can be obtained by eliminating the time from Eqs. (13.104) and (13.107). The trajectory equation is given by 
  ap 3 NA sin θ dh (h + 1)f3 + NG =− 2 f4 2 ac f1 dθ h (h + 2)2   exp(−κap h) + NA Nζ κap 1 ± exp(−κap h)   3 f2 (h + 1)2 + NG cos θ (13.108) − 2 where NA is the attraction number defined as NA =

AH ac2 9π µap4 U∞ Asph

(13.109)

Nζ is the electrostatic (repulsion) number defined as Nζ =

6π ǫζc ζp ap AH

(13.110)

and NG is a dimensionless gravity number defined as NG =

2ρgac2 9µU∞ Asph

(13.111)

In the previous section when dealing with the deposition rate due to an impinging jet, the Brownian motion was taken into consideration. Consequently, the gravitational parameter, Gr, double layer parameter, Dl, and adhesion parameter, Ad, were defined relative to the Brownian motion thermal energy kB T . As the diffusion mechanism is not considered here, the gravitational NG , electrostatic number Nζ , and the attraction number NA , cannot be defined relative to kB T . Consequently, even though these dimensionless numbers represent the strength of the forces to which they are named, their bases are different.

13.4

LAGRANGIAN APPROACH

503

Figure 13.19. Paths of a colloidal particle: trajectory a leads to a collision, trajectory b is the limiting trajectory, and trajectory c leads to no collision.

The electrostatic repulsion number Nζ represents the ratio of electrostatic repulsive to London–van der Waals attractive forces. The gravitational number NG represents the ratio of gravitational force to the viscous force modified by (ac /ap )2 . Similarly, the attraction number NA represents the ratio of the dispersion attraction force to the viscous force modified by (ac /ap )2 . Spielman and FitzPatrick (1973) in their analysis of colloidal deposition on spherical collectors neglected the term NG (ap /ac ), which appears on the right hand side of Eq. (13.108) and further assumed negligible electrostatic repulsion forces, i.e., Nζ = 0. On the other hand, Spielman and Cukor (1973) assumed NG = 0 but incorporated electric double-layer repulsion in their analysis. Figure 13.19 shows the paths of a colloidal particle near a spherical collector. Three types of particle trajectories are shown. Particle trajectory a leads to a collision. Trajectory b is the limiting trajectory separating a collision trajectory from a noncollision trajectory. Trajectory c is a non-collision trajectory. All particles within the stream tube T would lead to a collision. The capture efficiency is given by Eq. (13.5) as η=

2ψL ac2 U∞

(13.112)

For the case of (ac /ap ) ≪ 1, the stream function for the flow over a spherical collector is approximated by ψ=

3 Asph ac2 U∞ 4




ap ac

2

(h + 1)2 sin2 θ

and the collision efficiency becomes 3Asph η= 2




ap ac

2



lim (h + 1)2 sin2 θ

θ→0

(13.113)

Solution of Eq. (13.108) by backward integration from θ = π and h = 0 to large values of h with θ → 0 would then, in principle, determine the limit of (h + 1)2 sin2 θ.

504

DEPOSITION OF COLLOIDAL PARTICLES

Hence, the collision efficiency can be determined. In practice, however, there is an infinite attraction (NA = 0) at θ = π and h = 0. Spielman and FitzPatrick (1973) discussed the details of the initial condition necessary in order to be able to proceed with backward integration of Eq. (13.108). It should be recalled here that capture by pure interception for the case of a single isolated spherical collector is given by Eq. (13.7). For a collector within a packed bed of spheres, the corresponding capture efficiency is 3 ηI = Asph 2




ap ac

2

where, for an isolated spherical particle, Asph = 1. In the limit of a very large attraction number NA and in the absence of gravitational and electrostatic repulsion forces, Spielman and Goren (1970, 1971) and Goren and Fitzpatrick (1973) used limiting expressions for the universal hydrodynamic functions and obtained an analytical expression for the collision efficiency for a spherical collector as 
2 
  1/3 ap 3 4 9 η= Asph NA (13.114) 2 ac 3 5 for NA ≫ 1, with NG = Nζ = 0. The term appearing in the first square brackets of Eq. (13.114) is the pure interception capture efficiency for a spherical collector. Consequently, Eq. (13.114) can be written as 1/3  η 4 9 NA = (13.115) ηI 3 5 Clearly, for NA ≫ 1, η/ηI becomes larger than unity. The enhancement in the collection efficiency is due to the presence of the London–van der Waals attractive force. Similarly, Spielman and Goren (1970) obtained a limiting expression for the case of a cylindrical collector: 

η = 2Acyl




ap ac

2 


3π NA 4

1/3

(13.116)

for NA ≫ 1, NG = Nζ = 0. Here the term in the square brackets is the pure interception efficiency for a cylindrical collector. The hydrodynamic flow parameter Acyl is given by Acyl


 2ρac U∞ −1 1 2.0 − ln = 2 µ

(13.117)

13.4

LAGRANGIAN APPROACH

505

for an isolated cylinder (Lamb’s solution). For a fibre mat, the corresponding expression for Acyl is (Happel, 1959; Guzy et al., 1983; Adamczyk and van de Ven, 1981b)

Acyl



1  = − ln αc +  1 + αc2

−1

Happel’s model

(13.118)

where αc is the volume fraction of the cylindrical collectors in the fibre mat. The stream function for flow past a cylinder for the case of ap ≪ ac is defined as ψ = 2Acyl U∞ (r − ac )2

sin θ ac

(13.119)

Equation (13.119) is valid for Re ≤ 0.5. For the case of a cylindrical collector, ac becomes the radius of the cylinder. The attraction number for the case of the cylinder becomes AH ac2 NA = 9π µap4 U∞ Acyl and the ratio of the deposition efficiency to that of the interception efficiency for NA ≫ 1 becomes for the case of a cylindrical collector  1/3 η 3π = NA ηI 4

(13.120)

Once again, η/ηI becomes larger than unity for NA ≫ 1. For a spherical collector with Nζ = 0, the variation of the ratio of the capture efficiency to that due to pure interception as obtained by the numerical solution of Eq. (13.108) is shown in Figure 13.20. As would be expected, for a given value of the gravitational group NG , the capture efficiency ratio increases with the attraction number NA . For a fixed NA , the capture efficiency ratio increases with increasing gravitational number NG . The asymptotic solution given by Eq. (13.115) for Nζ = NG = 0 is also shown in Figure 13.20. Good agreement between the numerical and the asymptotic solution is evident for NA ≫ 1. Such a good agreement indicates that the separation distance s does not need to be very large compared to ap before the hydrodynamic interactions become weak (Probstein, 2003). For a cylindrical collector, Figure 13.21 shows the variation of the ratio of the capture efficiency to that of pure interception computed using an equation corresponding to Eq. (13.108) (Spielman and FitzPatrick, 1973). The asymptotic solution given by Eq. (13.116) for NA ≫ 1, NG = Nζ = 0 is also shown on Figure 13.21. Good agreement is once again present for NA > 1 between the two solutions. In the absence of gravitational force NG = 0, Spielman and Cukor (1973) numerically solved the trajectory equation for the case of a spherical collector for both constant potential and constant charge cases. Here the solution is governed by the scaled double layer thickness κap , the attraction (adhesion) number

506

DEPOSITION OF COLLOIDAL PARTICLES

Figure 13.20. Variation of normalized capture efficiency for a spherical collector with attraction number NA (Spielman and Fitzpatrick, 1973).

NA , and the repulsion number Nζ . The analysis is made using an unretarded dispersion potential. The expression for the repulsive double layer is restricted to |ζc ∼ ζp | < 0.05 V and κap > 10. A typical potential energy of interaction is shown in Figure 13.22. Depending on the values of κap , NA , and Nζ , four possible modes of capture are predicted. For the case of κap ≫ 1 and constant charge, Figure 13.23 shows the four modes of capture. In zone I, where no capture is possible, the particles are unable to surmount the repulsive energy barrier as shown in Figure 13.22. Here all particles

Figure 13.21. Variation of normalized capture efficiency with attraction number NA for capture of neutrally buoyant particles by a cylinder. Hydrodynamic interactions and van der Waals attraction are incorporated (Spielman and Fitzpatrick, 1973).

13.4

LAGRANGIAN APPROACH

507

Figure 13.22. Variation of combined London–van der Waals attraction and electric double layer repulsion potentials with separation distance between a colloidal particle and a collector.

escape capture. In zone II, capture occurs at the primary minimum. In this region, Nζ /κap is small indicating a weak repulsive energy barrier. In zone III, capture occurs at the secondary minimum. Here, no approaching particles are carried over the repulsive energy barrier. The secondary minimum is sufficiently deep to prevent some particles from escaping. In zone IV, a small region is present where combined capture by the primary and secondary minima occurs (Spielman, 1977). The effect of the dimensionless inverse Debye length κap on the location of the four zones is shown in Figure 13.24 for the case of constant surface charge and in Figure 13.25 for the case of constant surface potential.

Figure 13.23. Modes of capture of non-diffusing particles by a spherical collector at constant surface charge with κap ≫ 1. Hydrodynamic interactions, London–van der Waals attraction, and double layer repulsion are incorporated with NG = 0 (Spielman and Cukor, 1973).

508

DEPOSITION OF COLLOIDAL PARTICLES

Figure 13.24. Regions of capture modes for non-diffusing particles by a spherical collector at constant surface charge for selected values of κap for NG = 0 (Spielman and Cukor, 1973).

Figure 13.26 illustrates the variation of the capture efficiency (normalized with pure interception capture efficiency) with the attraction number at fixed repulsion numbers. Recall that the repulsion number Nζ represents the ratio of repulsive forces to attractive forces. In other words, for a constant non-zero value of Nζ , the ratio of the repulsive to the attractive forces is fixed. Consequently, for a given system, where the physical properties are held constant, a change in NA is derived from changes in the flow velocity U∞ . At Nζ = 0, the capture efficiency increases with NA as

Figure 13.25. Regions of capture modes for non-diffusing particles by a spherical collector at constant potential for selected values of κap for NG = 0 (Spielman and Cukor, 1973).

13.5

DEPOSITION EFFICIENCY AND SHERWOOD NUMBER

509

Figure 13.26. Normalized efficiency vs. attraction number at fixed selected repulsion numbers and κap = 10 for the spherical collector: —— constant charge; − − − constant potential (Spielman and Cukor, 1973).

a consequence of increasing the attractive forces. A high capture efficiency can be achieved either by increasing the value of the Hamaker constant or decreasing the flow velocity. All the curves for non-zero Nζ values give a lower capture efficiency. This is to be expected due to the presence of a repulsive force. Consider now a given value of Nζ , say Nζ = 200 for the constant charge case. As stated earlier, a change in NA can be attributed to a change in the fluid velocity which may be capable of propelling a particle over the repulsive energy barrier to be captured at the collector surface. Indeed, for Nζ = 200, the capture efficiency is slightly below that for Nζ = 0 at NA < 10−5 . However, as NA increases, it leads to a weaker flow which may not be able to push particles over the repulsive energy barrier. Consequently, increasing NA leads to a lower capture and, finally, when NA is sufficiently large, no capture occurs.

13.5

DEPOSITION EFFICIENCY AND SHERWOOD NUMBER

The deposition rates were reported in terms of Sherwood numbers when the analysis was made using the Eulerian approach. However, capture or deposition efficiency, η, was used when dealing with the Lagrangian approach. The Sherwood numbers for the case of the spherical collector reported in Section 13.3 can be cast in terms of the capture efficiency. The establishment of a relationship between the capture efficiency and the Sherwood number for the case of a spherical collector is given below. Recall that the average Sherwood number was defined (see Table 13.3 ) as 1 Sh = S


S

j⊥ ac D ∞ n∞



ac dS = SD∞ n∞



S

j⊥ dS

(13.121)

510

DEPOSITION OF COLLOIDAL PARTICLES

The integral



S j⊥ dS

is given by 

S

j⊥ dS = 2π ψL n∞

(13.122)

where 2π ψL n∞ represents the total flux of the particles. Here ψL represents the limiting stream function for a given capture mechanism. Combining Eqs. (13.121) and (13.122) leads to Sh =




ac SD∞ n∞



(2π ψL n∞ )

(13.123)

From the definition of the capture efficiency for the case of a spherical collector as given by Eq. (13.5) and by setting S = 4π ac2 , Eq. (13.123) becomes Sh =




ac 2 4π ac D∞ n∞



(ηπ ac2 U∞ n∞ )

leading to ηP e (13.124) 4 where P e = ac U∞ /D∞ for a spherical collector. Equation (13.124) relates the Sherwood number of a spherical collector to its capture efficiency. The equivalent expression for the case of a cylindrical collector is given as Sh =

Sh =

ηP e π

Using Eqs. (13.124), (13.44), and (13.47), one can obtain an expression for the deposition efficiency for purely diffusive (Brownian) transport, ηD , around an isolated spherical collector as ηD = 4Sh/P e = 4[0.624P e−2/3 ]

(13.125)

When the deposition is dominated by gravitational force, using Eqs. (13.92), (13.124), and the Stokes–Einstein equation, (12.13), together with Eq. (12.7) we can define a deposition efficiency due to gravity, ηG , as ηG =

2 ρgap2 9 µU∞

(13.126)

Finally, the deposition efficiency due to interception is given by Eq. (13.7) 3 ηI = 2




ap ac

2

(13.127)

13.5

DEPOSITION EFFICIENCY AND SHERWOOD NUMBER

511

An overall deposition efficiency, η0 , can be obtained by adding the deposition efficiencies due to diffusion, gravity and interception η0 = ηD + ηG + ηI

(13.128)

Such correlations were developed for filtration by Yao et al. (1971), who based their work on earlier developments in aerosol filtration (Friedlander, 1958; Fuchs, 1964). Subsequently, correlations for the deposition efficiency were modified to include the effects of the van der Waals attraction (Rajagopalan and Tien, 1976). A more complete description of these correlations can be found in Elimelech et al. (1995). One should note, however, that the simple additivity of the different mechanisms for deposition is an approximation, and although it is found to be quite accurate for most applications, such additivity of different mechanisms may not be fundamentally sound. Furthermore, these correlations do not account for the influence of electric double layer interactions on the deposition efficiency. Thus, the numerical methodologies outlined in the previous sections are preferable when more accurate results are desired. The correlation given by Eq. (13.128) predicts the dependence of the deposition efficiency on the particle size, and provides a means for assessing the dominant mechanisms for the deposition process. Figure 13.27 depicts the variation of the overall deposition efficiency with particle radius obtained using Eq. (13.128), as well as the individual contributions due to diffusion, gravity, and interception. It is interesting to note that there is a minimum in the deposition efficiency observed for a

Figure 13.27. Dependence of the deposition efficiency on particle radius. The solid line shows the overall deposition efficiency, while the other lines depict the contributions due to diffusion, interception, and gravity. U∞ = 2.0 × 10−3 m/s, ac = 0.1 mm; ρp = 1100 kg/m3 ; ρ = 1000 kg/m3 and T = 298 K.

512

DEPOSITION OF COLLOIDAL PARTICLES

particle radius of about 0.6–0.7 µm. Particles smaller than ca. 0.2 µm exhibit purely Brownian (diffusion controlled) deposition, while for ap > 1.5 µm, the deposition process is governed solely by interception and gravity. The particle size dependence of η0 shown in Figure 13.27 is only applicable in absence of electric double layer interactions. In presence of other interactions, the deposition efficiency changes quite markedly. However, the presence of a minimum deposition efficiency corresponding to a particle radius of 0.5–1.0 µm is almost universally observed. This size dependence is of immense importance in filtration, since the behavior shown in Figure 13.27 indicates that the filtration will be least efficient (capture efficiency will be lowest) for particle radii in the range of 0.5 to 1.0 µm.

13.6

EXPERIMENTAL VERIFICATIONS

In the previous sections we discussed the theoretical development of the deposition of colloidal particles on a surface. The mass transfer rate was given by dimensionless groups, either in terms of a collection efficiency η or in terms of a Sherwood number Sh. For the case of a spherical collector, the relationship between η and Sh was derived and it is given by Eq. (13.124). For the case of deposition in a stagnation flow, spinning disc, and channel flow, the theoretical descriptions of the flow field are well established. On the other hand, for the case of flow in a packed bed or in a fibre mat, the flow field is deduced using approximate flow models. Consequently, any disagreement between the experimental and theoretical deposition rates in the case of a well-defined flow geometry can only be attributed to the manner by which interparticle forces are modelled. Hence, experimental data from well-defined flow geometries are valuable for establishing the validity of the DLVO theory and deposition models. Hull and Kitchener (1969) studied the deposition of polystyrene particles (ap = 0.154 × 10−6 m) onto a rotating disc under laminar flow conditions. In their first set of deposition experiments, the surface of the rotating disc was covered with polyvinylpyridine copolymer film giving a zeta potential of 0.072 V. The polystyrene particles zeta potential was −0.070 V. For this case of opposite signs in the zeta potential, the experimentally measured deposition rate was very close to that given by the Levich theory, where 2/3 j⊥ = 0.62D∞


1/6 ρ ω1/2 n∞ µ

(13.129)

The Levich equation represents the case where the deposition process is purely convection-diffusion (mass-transfer) controlled in the absence of interparticle forces. The agreement of the experimental results with the Levich theory is not surprising as, for the system studied, the adhesion number Ad is of order 10−1 and the double-layer Dl is negative. For the case of a stagnation flow, Figure 13.9 showed that when Dl is small or negative, the Sherwood number for a given flow Reynolds number is not

13.6

EXPERIMENTAL VERIFICATIONS

513

sensitive to variations in Dl. Equation (13.129) gives j⊥ 2/3 = 0.62D∞ n∞ ω1/2


1/6 ρ µ

(13.130)

The right side of Eq. (13.130) is a constant for a given system. Making use of the definitions given in Table 13.3, Eq. (13.130) leads to Sh = 0.62 Sc1/3 Re1/3

(13.131)

In this case, the local and the average Sherwood numbers are the same. For negative and small values of Dl, the curves of Figure 13.9 conform to the relationship given by Eq. (13.131) and in turn to Eq. (13.130). According to Hull and Kitchener (1969) experimental measurements for the case of opposite signs in zeta potential, the particles flux is given as j⊥ = 7.59 × 10−8 n∞ ω1/2 (in m/s1/2 ) whereas the Levich theory gives 7.66 × 10−8 m/s1/2 . The agreement between the experimental measurements and the Levich theory is excellent. Hull and Kitchener carried out further experimental tests where the zeta potential of the surface of the rotating disc is rendered negative by coating the disc surface with Formvar. Here both the particles and the disc surface have the same signs for the zeta potential. In this case, there is an electrostatic repulsion between the polystyrene particles and the disc surface. The experimental results showed poor agreement with the deposition theory which takes into account the dispersion and electrostatic forces. Clint et al. (1973) carried out similar experiments to those conducted by Hull and Kitchener. Clint et al. regulated the zeta potential of the polystyrene particles (ap = 1 − 0.214 × 10−6 m) and the rotating disc surface by the addition of Ba(NO3 )2 . The zeta potential of the particles varied from about −0.025 to −0.012 V and the deposition surface zeta potential from −0.005 to −0.007 V by changing the electrolyte concentration from 5 to 20 mol/m3 , respectively. When Clint et al. (1973) accounted for the formation of doublets and triplets within the bulk suspension for the case of high electrolyte concentration, they were able to obtain reasonable agreement with the deposition theory where dispersion and electrostatic forces are accounted for. However, poor agreement with theory was obtained at low electrolyte concentrations where the repulsive forces are strongest. Clint et al. pointed out that the deposition rate was very sensitive to the value used for the zeta potential of the deposition surface. Changing ζc from −0.0061 to −0.0062 V had the effect of changing the deposition rate by 16%. The change in ζc corresponds to 0.1 mV which is well within experimental error for a zeta potential measurement. Bowen and Epstein (1979) carried out deposition experiments from a flowing suspension of silica particles in a parallel-plates channel. The deposition corresponding to their experimental Run II-2 is shown in Figure 13.28. Table 13.4 gives details

514

DEPOSITION OF COLLOIDAL PARTICLES

Figure 13.28. Deposition onto a positive 2VP/S substrate in a parallel plate channel. Run II-2 (Bowen, 1978).

of their experimental run. For this test, the silica particles and the deposition surface have opposite signs for the zeta potential. It is clear that the initial variation of the concentration of deposited particles with time is linear and the deposition rate declines with time. The experimental initial deposition rate was found to be 7.87 × 106 m−2 s−1 . The theoretical value as calculated by Adamczyk and van de Ven (1981b) is given as 7.63 × 106 m−2 s−1 . Again excellent agreement between the experimental and theoretical deposition rates exists. As in the case of the rotating disc deposition TABLE 13.4. Physical Data for Run II-2 of Bowen (1978) and Bowen and Epstein (1979). Silica particle radius, ap Zeta potential of particle, ζp Zeta potential of deposition surface, ζc Temperature, T Axial distance, x Channel half-width, b Particle number concentration, n∞ Average axial velocity, Vm Counterion concentration Hamaker constant (assumed), AH Diffusion coefficient, D∞ κap Dl = 4π ǫap ζp ζc /kB T Ad = AH /6kB T P e = 3Vm ap3 /2b2 D∞ Channel Reynolds number, 4Vm bρ/µ

0.324 ×10−6 m −0.075 V 0.015 V 298 K 0.125 m 4.26 ×10−4 m 0.795 ×1014 m−3 0.149 m/s 6.8 ×10−6 mol/L 1 × 10−21 J 7.53 ×10−13 m2 /s 2.77 −780 0.04 0.056 284

13.6

EXPERIMENTAL VERIFICATIONS

515

measurements, for the case of opposite signs for the zeta potential, the deposition rate is good in agreement with Levich-type analysis. For the case of slit channel flow, the theoretical analysis was conducted by Levesque (1928) (see Adamczyk and van de Ven, 1981a,b; Bowen et al., 1976). The Levesque-type local deposition rate in a channel is given by
1/3 2 n∞ D∞ (13.132) j⊥ = bŴ(4/3) 9γ where γ =

2D∞ x 3Vm b2

(13.133)

and D∞ =

kB T 6π µap

(13.134)

Here Vm is the average velocity in the channel, b is the channel half-width, x is the axial downstream distance at which the deposition measurements are taken, and Ŵ(4/3) is the Gamma function given as 0.893. The Levich deposition rate is given as 7.22 × 106 m−2 s−1 which agrees quite well with the experimentally determined value. In dimensionless form, Eq. (13.132) is given by
1/3 ap j⊥ Pe Shl = = 0.678 − (13.135) n∞ D ∞ x where

x b The Peclet number is given after Adamczyk and van de Ven (1981a) as −

x=

Pe =

3Vm ap3 2b2 D∞

(13.136)

(13.137)

Similar to the previous studies with a rotating disc, poor agreement for deposition was obtained when the deposition channel wall had a zeta potential ζc of the same sign as the colloidal particles. ζc varied from −0.009 V to −0.046 V. The theoretical analysis predicted no deposition whereas experimentally significant deposition was observed. In summary, when the electrostatic forces are attractive, good agreement was observed between the theoretical analysis and the experimental measurements. Hence, the mass transfer is not much affected by either the attractive dispersion or the attractive electrostatic forces. In most cases, Levich-type analysis predicts the deposition rate quite adequately. However, when strong repulsive forces due to the electric double layer are present, large deviations between the theoretical analysis and the experimental measurements occur. There are many possible explanations for these discrepancies. As was indicated earlier, when electrostatic repulsive forces are present, the deposition

516

DEPOSITION OF COLLOIDAL PARTICLES

rates become very sensitive to the exact value of the zeta potential used. Moreover, unlike an idealized surface, real surfaces have small-scale roughness elements and a non-uniform distribution of charge sites on the surface. The interaction between a given particle and a substrate is strongly dependent upon the nature of the local areas coming into contact (Bowen and Epstein, 1979). Kihira et al. (1992) were able to develop a mathematical model that accounts for the discreteness of the surface charge where it was assumed that it followed a Gaussian distribution with an assigned mean and a standard deviation. The mean Stern potential is assigned to the surface zeta potential. The standard deviation becomes a measure of the heterogeneity of the surface. By adjusting the standard deviation of the surface potential, they were able to match their experimental coagulation stability ratio with their theoretical model. To this end, a new theory of electric double layer interactions is required that can account for heterogeneities of the geometry and the charge distribution as applied to real surfaces. Sanders et al. (2003) conducted stagnation point flow deposition experiments with silane-treated (hydrophobic) silica particles (ap = 0.47 µm) onto silane-treated (hydrophobic) collectors at a pH of 4 for different electrolyte (NaCl) concentrations. As shown in Figure 13.29, the experimental deposition data lie much above the DLVO predictions calculated for 0.1 M NaCl solution. The particle deposition is nearly independent of NaCl concentration and appears to depend predominantly upon the flow rate as characterized by the jet Reynolds number, see Table 13.3. Their data suggest that an additional attractive particle-collector interaction that is not

Figure 13.29. Dimensionless mass transfer (expressed as Sherwood number) as a function of NaCl concentration and Reynolds number for silane-treated (hydrophobic) silica particle deposition onto a silane-treated (hydrophobic) collector: ap = 47 µm; pH = 4; Ad = 1.75; Gr = 0.187. Solid line shows mass transfer rates predicted using DLVO theory for 0.1 M NaCl. The zeta potential of silane-treated silica are −38, −21, and −11 mV at 0.001, 0.01, and 0.1 M NaCl, respectively (Sanders et al., 2003).

13.6

EXPERIMENTAL VERIFICATIONS

517

Figure 13.30. Effect of collector surface properties on the rate of bitumen droplet deposition due to jet impingement (Sanders et al., 2003).

included in the DLVO analysis, exists. This observation is in agreement with other studies of hydrophobic interactions (Israelachvili, 1992; Israelachvili and Pashley, 1984; Rabinovich and Yoon, 1994; Song, et al., 2000) and is also in agreement with more specific studies of hydrophobic silica particles (Xu and Yoon, 1989, 1990). Sanders et al. (1995, 2003) also conducted deposition experiments using bitumen droplets on silane-treated, non-treated, and bitumen coated impingement plates. They found excellent agreement with the DLVO theory while using the non-treated hydrophilic impingement plate. However, a lower deposition rate than theoretically predicted by the DLVO theory was found for the silane-treated hydrophobic plate and bitumen-coated plate. Their findings are shown in Figure 13.30. It is clear that in deposition experiments, not only the surface charge can play a role in modifying the deposition rate, but also the nature of the surface hydrophobicity as well as any additional steric interaction. For the case of deposition in packed beds, FitzPatrick and Spielman (1973) conducted a very extensive experimental study for the deposition of latex particles in beds of glass spheres. The colloidal particles had a wide range of size with ap being in the range of 0.36–11 × 10−3 µm and the collector size ac being in the range of 0.05–2 mm. For the case of a negligible double-layer repulsion, i.e., small Nζ numbers and negligible gravity numbers NG , they plotted the dimensionless filter coefficient λ/λI versus the attraction number NA as shown in Figure 13.31. The filter coefficient λ is defined by Eq. (13.15) and the pure interception filter coefficient λI is given by λI =

9αc Asph ap2 8ac3

(13.138)

The flow model correction factor Asph used in the attraction number NA is due to Happel (1958) and it is given by Eq. (13.10). The solid line on Figure 13.31 is from the

518

DEPOSITION OF COLLOIDAL PARTICLES

Figure 13.31. Variation of the dimensionless filter coefficient λ/λI with the attraction number for negligible repulsion and sedimentation. Solid line: Theoretical analysis with Nζ = NG = 0 (adapted from Fitzpatrick and Spielman, 1973). The dimensionless filter coefficient is given by λ/λI = 8ac3 λ/(9αc Asph ap2 ) and the attraction number is given by NA = AH ac2 /(9π µap4 U∞ Asph ).

solution of Eq. (13.108). Good agreement between the experimental and theoretical deposition results is evident even though to some extent the theoretical analysis is dependent on the choice of the flow model Asph . When repulsion forces are not negligible, considerable scatter in η/ηI versus NA plot was observed and the data could not be correlated with the repulsion number Nζ . The effect of sedimentation on λ/λI is shown in Figure 13.32 for NG = 4.3, 12.9, and 43.2. Curves for NG = 0 are also shown for comparison purposes. The solid curves are due to theory, Eq. (13.108). It is clear that there is good agreement between the theory and the experimental data. The dashed lines give the dimensionless filter coefficient for the combined interception and gravity. They are given by λ 2 = 1 + NG λI 3

for NG > −3/2

(13.139)

The development of the above equation does not include the hydrodynamic interaction between a colloidal particle sand the collector surface. At higher NG numbers, the exact solution and that given by Eq. (13.139) become closer. Vaidyanathan and Tien (1988) re-examined FitzPatrick’s (1972) data by incorporating non-uniform potentials in the trajectory analysis. The average collection efficiency is defined as η¯ =



∞

−∞



∞

−∞

fp (ζp ) fc (ζc ) η(ζp , ζc ) dζp dζc

(13.140)

13.6

EXPERIMENTAL VERIFICATIONS

519

Figure 13.32. Variation of the dimensionless filter coefficient with attraction number for different gravity numbers. Solid line: theoretical analysis; dashed line: Eq. (13.139) (Spielman and Fitzpatrick, 1973). The dimensionless filter coefficient is given by λ/λI = 8ac3 λ/(9αc Asph ap2 ) and the attraction number is given by NA = AH ac2 /(9π µap4 U∞ Asph ).

where fc (ζc ) and fp (ζp ) are probability density functions of the collector and particle zeta potential, respectively. η(ζp , ζc ) is the collection efficiency when the collector has a zeta potential ζc and the particle ζp . Figure 13.33 shows the filter coefficient variation with the ionic strength. The collector and particle zeta potentials are included

520

DEPOSITION OF COLLOIDAL PARTICLES

Figure 13.33. Filter coefficient variation with electrolyte ionic strength (Fitzpatrick and Spielman, 1973). Non-uniform potentials type solution from Vaidyanathan and Tien (1988).

in the plot. The agreement between the prediction of Vaidyanathan and Tien with the experimental data of FitzPatrick (1972) for the filter coefficient λ is found to be reasonably good at the onset of unfavorable surface interactions but poor as the ionic concentrations decline further. Guzy et al. (1983) showed that particle retention on unconsolidated fibrous porous media depends on the flow model used in estimating the flow field within the porous bed. The various flow models, e.g., the Happel, the Kuwabara, and the Brinkman cell models, lead to small but very significant differences in the streamlines near the collector surface. The differences are most pronounced for lower porosity beds of filters. The sensitivity of the collision efficiency to the flow model and electric doublelayer thickness was illustrated by Guzy et al. (1983). Figure 13.34 shows the sum of the London–van her Waals attractive force FA and the electric double layer repulsive − force FR for Dl = 100, Ad = 1.0, and λ = 0.4 with Da = Gr = 0. Plots for various κap ranging from 2 to 80 are shown. The total interaction force has a repulsive peak corresponding to all values of κap shown except for κap = 80. As κap is increased, the location of the repulsive peak shifts closer to the collector surface (h → 0). The magnitude of the peak first increases with increasing κap , and then decreases for κap > 30. It is of interest to compare the force profiles corresponding to κap = 5 and 50 (also κap = 2 and 57), since the maxima of the force curves are quite similar (Fz∗ ∼ 140 for κap = 5 and 50, while Fz∗ ∼ 70 for κap = 2 and 57). However, the maximum values occur at different separation gaps, h. Due to the lubrication effects, the “attractive” hydrodynamic force is weaker near the collector surface than farther out. Thus, for an expanded double layer, κap = 2, the hydrodynamic force is sufficient to move the particle through the energy barrier as opposed to the case of κap = 57. A small change in the value of the normal velocity near the collector surface (i.e., the position of the streamline) can make a difference as to whether capture will occur

13.7

APPLICATION OF DEPOSITION THEORY

521

Figure 13.34. Variation of the sum of the electric double layer and London–van der Waals dimensionless forces as given by Eq. (13.76) with dimensionless gap length.

or not. This is why an accurate prediction of the flow field is crucial in determination of the deposition efficiency. It is clear from the comparison between the experimental measurements and the theoretical analysis that there is much room for improvement in the DLVO theory as applied to real surfaces and in the flow models for deposition in filter beds (Hirtzel and Rajagopalan, 1985). This is particularly important for repulsive electric double layer interactions and surface hydrophobicity, for which case, a consistent deviation between the theory and the experiments have been reported. For a comprehensive treatise on granular filtration and deposition of particles, the reader is referred to Tien (1989) and Elimelech et al. (1995). 13.7

APPLICATION OF DEPOSITION THEORY

Theoretical analysis of particle deposition is a mature subject, and the foregoing discussion in this chapter provides a summary of the fundamental aspects of the theory. The primary driving factors for studies on particle deposition are the immense importance of the subject in a variety of industrial and environmental applications. Although the basic modeling approach still remains firmly rooted in the Eulerian or trajectory analysis, several modifications in the theoretical construct are needed before one can apply these models to predict deposition processes in real systems. Here we summarize some of the key directions. 13.7.1 Deposition in Porous Media Hydrodynamic Dispersion in Porous Media The Eulerian approach presented earlier considered diffusion of the particles. However, we noted in Chapter 8 that

522

DEPOSITION OF COLLOIDAL PARTICLES

hydrodynamic dispersion in narrow channels can cause considerable variations in the solute (particle) distribution. A straightforward modification of the particle transport equation in porous beds of collectors involves replacing the diffusion coefficient with the hydrodynamic dispersion coefficient. The hydrodynamic dispersion coefficient, Dh , for a one dimensional flow is given by Dh = D∞ + αL Vp

(13.141)

where D∞ is the Stokes–Einstein diffusivity of the particle, αL is the longitudinal dispersivity parameter, and Vp is the interstitial velocity, which is the average velocity of the fluid through the packed bed of collectors. Long Term Deposition Behavior The deposition efficiencies or Sherwood numbers discussed in the previous sections are all based on the assumption that the deposited particles do not accumulate on the collector, and that the collector surface is clean. Such an assumption applies only during the initial stages of deposition onto a clean collector. Consequently, the deposition efficiency provided by the theoretical analysis discussed earlier is applicable to the initial deposition behavior, and is generally termed as the initial deposition efficiency. In reality, as the collector surface becomes covered with deposited particles, its capture efficiency changes with the extent of particle coverage. In context of filtration processes, such a transient behavior is more pertinent. There can be two outcomes of the transient variations of the deposition efficiency: When the deposited particles hinder further deposition, the process is referred to as blocking, while if the deposited particles enhance further deposition, the process is called ripening. The transient deposition behavior is typically observed in deep bed filtration and in chromatographic separations. Consider a packed column of collectors, through which a particle suspension is continuously injected. Assume the inlet particle concentration to be C0 . The effluent particle concentration, Ce , exhibits variations with time in such a system. The transient variation of the concentration ratio Ce /C0 is generally referred to as particle breakthrough curves and is depicted in Figure 13.35. The initial stages of the particle breakthrough curves can be represented in terms of the deposition theories applicable to clean beds. However, modeling the long term behavior of these breakthrough curves owing to blocking or ripening necessitates additional insights into the role played by previously deposited particles in modifying the collector efficiency. The basic physics of blocking involves reduction in the available surface sites on the collector grains due to the deposited particles. Considering the fact that similarly charged particles will electrostatically repel each other, it is discernable that blocking is engendered by (i) physically reducing the available collector surface area (steric hindrance), and (ii) by rendering the collector surface less attractive or less favorable (electrostatic hindrance). Ripening, on the other hand, is predominantly caused by increase in the net capture cross section of the collectors, resulting in a greater interception of flowing particles. The dynamics of collector surface blocking by deposited particles has received significant attention over the past several decades, and one of the most prominent

13.7

APPLICATION OF DEPOSITION THEORY

523

Figure 13.35. Schematic representation of the variation of particle breakthrough curves with time during blocking and ripening.

developments in this regard is the random sequential adsorption (RSA) model (Feder, 1980; Schaaf and Talbot, 1989; Adamczyk et al., 1999; Ko et al., 2000). The RSA model provides a relationship between the deposition efficiency and the extent of collector surface coverage by deposited particles in the form of a chemical kinetic equation. Solving the particle transport model in conjunction with the RSA based kinetic equation provides a reasonable mathematical construct for analyzing long term deposition behavior (Ko et al., 2000). Release of Deposited Particles Although release of deposited particles has received considerably less attention, it is observed in nature, and has tremendous consequences. Colloid mobilization in groundwater is a consequence of release of deposited particles from the soil matrix. Due to chemical variations in groundwater (such as a sudden change of salt concentration or pH following infiltration of fresh rain water), the electrostatic interactions between the particles and collectors can be dramatically modified. Such a modification can result in release and mobilization of colloidal particles. Empirical approaches based on kinetic expressions are available to assess the extent of release in porous media (see, for instance, Bhattacharjee et al., 2002). Chemical Heterogeneity of the Collector Bed As discussed earlier, chemical heterogeneity of collector beds can have a profound influence on the deposition behavior. Incorporation of such effects in models for predicting the performance of packed bed filters can be quite challenging if one has to consider the exact heterogeneous nature of the collectors. An effective means for modeling such heterogeneities in a simplified manner for packed beds is afforded by the two-site patchwise charge heterogeneity model (Johnson et al., 1996). This model assumes that the total deposition surface area of the collectors in a packed bed is composed of a favorable fraction, λf = Af /Ac and an unfavorable fraction, (1 − λf ), where Af is the favorable (attractive) surface area of the collectors, and Ac is the total collector surface area.

524

DEPOSITION OF COLLOIDAL PARTICLES

13.7.2 Colloid Transport Models in Porous Media The different mechanisms affecting the long term deposition process discussed above can be simultaneously incorporated in a theoretical formulation to assess the particle transport and breakthrough behaviors through porous media. Referring back to Figure 13.3, the general equation for particle transport in presence of hydrodynamic dispersion and blocking effects can be written (for one-dimensional transport through a packed bed of collectors) as (Ko et al., 2000) ∂ 2c ∂c f ∂θ ∂c = Dh 2 − Vp − ∂t ∂x ∂x π ap2 ∂t

(13.142)

Here, c is the local particle concentration in the flowing suspension at the location x, Dh is the hydrodynamic dispersion coefficient, Vp is the interstitial (pore) velocity, which can be related to the external (superficial) velocity, U∞ , of the injected suspension as Vp = U∞ /(1 − αc ). The term f is the collector surface area per unit length of the packed bed, ap is the depositing particle radius, and θ is the fraction of the collector surface covered by deposited particles at a given instant. For irreversible monolayer deposition, the fractional surface coverage is given by a kinetic expression of the form ∂θ = π ap2 kdep B(θ )c (13.143) ∂t Equation (13.143) is a kinetic expression that is applied as a source/sink term in the convection–dispersion equation (13.142). Here, kdep is the deposition rate constant, which for spherical collectors, can be related to the collector deposition efficiency through ηU∞ (13.144) kdep = 4 The term B(θ ) in Eq. (13.143) is the dynamic blocking function or the available surface function, which relates the modification of the deposition kinetics with the fractional surface coverage. One can express the dynamic blocking function as a virial of the form B(θ ) = 1 + a1 θ + a2 θ 2 + a3 θ 3 + · · ·

(13.145)

where the coefficients a1 , a2 , a3 , etc., are dictated by different models of adsorption. For simple Langmuirian adsorption, the blocking function is given by B(θ ) = 1 − θ. The Langmuirian blocking model is applicable to infinitesimal particles. For finite sized particles, one needs to consider the steric effects due to the occluded area by each particle. For hard spherical particles depositing on a planar collector, the random sequential adsorption (RSA) model provides the following values of the coefficients (Schaaf and Talbot, 1989): a1 = −4;

√ 6 3 a2 = ; π

and a3 =




176 40 √ − 2 3π π 3



13.7

APPLICATION OF DEPOSITION THEORY

525

It is interesting to note from the first virial coefficient that a hard sphere occupying a surface will occlude (prevent other particles to deposit on) four times its own projected area due to its finite size. The hard-sphere RSA model also provides insight about the maximum surface coverage on a collector beyond which no more particles can deposit. This maximum coverage is given by θ∞ = 0.546, which implies that if 54.6% of a collector surface is covered by hard spheres of finite dimensions, it can no longer accommodate any more particles. The RSA model has been explored in context of particle deposition studies to investigate how electric double layer repulsion between the adsorbed and depositing particles as well as hydrodynamic interactions modify the virial coefficients a1 to a3 in Eq. (13.145). Extensive discussions of these effects are available elsewhere (Adamczyk et al., 1994; Adamczyk et al., 1999; Ko et al., 2000). Here we provide a qualitative summary of the pertinent effects. Consider the deposition process on a planar collector for three possible situations, as shown in Figure 13.36. In each case, we consider the approach of a suspended particle near an already deposited particle on the planar surface. First, when one considers deposition of hard spheres in a quiescent fluid (case a), each sphere will block four times its projected area of the surface. When the particles are charged, as in case b, the projected area of the surface blocked will be larger, owing to the presence of the electric double layer around the particle. The double layer will effectively increase the deposited particle size, and consequently, the overall blocked area. Finally, in case (c), we observe the coupled influence of the electric double layer and a tangential flow field on the blocked area. Here, the combined influence of the electrostatic and hydrodynamic interactions results in an

Figure 13.36. Schematic representation of influence of electric double layer and hydrodynamic interactions on the dynamic blocking function. (a) Hard spheres in a quiescent medium; (b) charged hard spheres in a quiescent medium; (c) charged hard spheres in a tangentially flowing suspension. The regions on the planar substrate marked as blocked surfaces are regions where other particles cannot deposit. The shapes of the blocked regions vary owing to the different interactions.

526

DEPOSITION OF COLLOIDAL PARTICLES

elongated “shadow region” behind each deposited particle. It is therefore evident that the particle-particle electrostatic and hydrodynamic interactions can significantly modify the long term deposition kinetics as well as the maximum surface coverage of the collector. Incorporation of particle release and chemical heterogeneity on the collector surface in the filtration models involves further modification of the source/sink term of the convection-dispersion equation. A general format for including these effects is to consider the deposition on two types of surface sites, namely, favorable sites given by the fraction θf and an unfavorable fraction, θu . The total fractional coverage of the collector at any given instant can then be represented in light of a patchwise heterogeneity model as (13.146)

θ = λf θf + (1 − λf )θu

The corresponding rate expression for the patchwise deposition model can be written as (Chen et al., 2001; Bhattacharjee et al., 2002) ∂θf ∂θu ∂θ = λf + (1 − λf ) ∂t ∂t ∂t

(13.147)

For each type of site, the deposition and release kinetics can be modeled as ∂θf = π ap2 kf B(θf )c − krelease,f θf ∂t ∂θu = π ap2 ku B(θu )c − krelease,u θu ∂t

(13.148-a) (13.148-b)

In the above equations, kf and ku are the deposition rate constants on the favorable and unfavorable surface sites, respectively. Similarly, krelease,f and krelease,u represent the release rate coefficients from the favorable and unfavorable sites, respectively. While the deposition rate constants can be evaluated theoretically using the deposition efficiency, no theoretical model is available for the release rate constants. Once the transport model is formulated for a particular deposition problem, it can be solved for a given system employing appropriate boundary conditions. The application of the one dimensional and two dimensional versions of the model in assessing colloid transport in porous media and virus transport in groundwater have been discussed by Ko et al. (2000); Bhattacharjee et al. (2002); and Loveland et al. (2003). Such models can either be used as predictive tools for assessing particle deposition dynamics in packed media, or as tools for solving inverse problems for estimation of the deposition and release rate constants from experimental observations of colloid transport in porous media and aquifer beds.

13.8

13.8

SUMMARY OF DIMENSIONLESS GROUPS

527

SUMMARY OF DIMENSIONLESS GROUPS

13.8.1 Dimensionless Groups in the Flux Equation Double-layer parameter I, Dl: Measure of electrostatic to Brownian motion forces 4π ǫap ζp ζc kB T

Dl =

Double-layer parameter II, Da: Measure of asymmetry of the electrical double layer (ζc − ζp )2 2ζc ζp

Da =

Adhesion parameter, Ad: Measure of London–van der Waals to Brownian motion forces AH Ad = 6kB T −

Retardation parameter, λ: Measure of retardation due to the finite speed of light −

where λ = 10−7 m

λ = λ/ap

Gravitational parameter, Gr: Measure of gravitational to Brownian motion forces Gr =

2 ρgap3 9 µD∞

Double-layer thickness parameter, κap : Measure of electric double-layer thickness κap =




2e2 z2 n∞ ǫkB T

1/2

ap =




2000e2 z2 NA M ǫkB T

1/2

ap

Table 13.3 gives the definitions of the Reynolds, the Peclet, and the Sherwood numbers for the various geometries. For the case of a cylinder, the dimensionless groups in Table 13.3 are the same as for the case of a spherical collector with ac being the cylindrical collector radius. 13.8.2 Dimensionless Groups in the Trajectory Equation Electrostatic number, Nζ : Measure of electrostatic repulsion to London–van der Waals forces Nζ =

6π ǫζc ζp ap AH

528

DEPOSITION OF COLLOIDAL PARTICLES

Attraction number, NA : Measure of London–van der Waals to viscous forces NA =

AH ac2 9π µap4 U∞ Ac

where the collector area, Ac is given by Ac ≡ Acyl for a cylindrical collector and Ac ≡ Asph for a spherical collector Gravity number, NG : Measure of gravitational to viscous forces NG =

2ρgac2 9µU∞ Ac

Double-layer thickness parameter, κap : Measure of electric double-layer thickness

2 2 1/2 1/2 2000e2 z2 NA M 2e z n∞ ap = ap κap = ǫkB T ǫkB T 13.9

NOMENCLATURE

a a1 , a2 , a3 ac ap AH Ad Ab Ac Acyl Asph B(θ ) c dc dp d d⊥

spherical particle radius, m coefficients of the dynamic blocking function collector radius (spherical or cylindrical), m colloidal particle radius, m Hamaker constant, J adhesion parameter, AH /6kB T empty bed cross-sectional area, m2 dimensionless flow parameter (= Acyl or Asph ) dimensionless flow parameter accounting for the presence of cylindrical particle in a fibre mat, Eqs. (13.117) and (13.118) dimensionless flow parameter accounting for the presence of spherical particle in a suspension, Eq. (13.10) dynamic blocking function or available surface function particle concentration, m−3 or mol/m3 spherical collector diameter colloidal particle diameter correction factor for diffusion coefficient parallel to surfaces, dimensionless correction factor for diffusion coefficient normal to surfaces, dimensionless

13.9

Da Dl Drr Drz Dzz D1∞ , D2∞ =

D f1 , f2 , f3 , f4 Fr Fz Fz∗ Fhyd Fgr Fgz FA FR F g Gr h i j jr j⊥ kB kdep kf , k u L mp n nc no n∞ NA NG Nζ Pe Q r Re

NOMENCLATURE

529

double layer parameter due to potential asymmetry, (ζc − ζp )2 /2ζc ζp double layer parameter, 4π ǫap ζc ζp /kB T diffusion coefficient correction for parallel to surfaces, m2 /s diffusion coefficient correction for normal and tangential movements, m2 /s diffusion coefficient correction for movements normal to surfaces, m2 /s particle diffusion coefficient in infinite dilution, m2 /s, given by kB T /6π µa diffusion coefficient tensor, m2 /s universal hydrodynamic correction functions (Tables 13.1 and 13.2), dimensionless force in r-direction, N force in z-direction, N dimensionless interparticle z-directed force hydrodynamic forces, N gravitational force in r-direction, N gravitational force in z-direction, N dispersion force (attractive), N electrostatic (repulsion) force, N fluctuating force vector, N gravitational acceleration, m/s2 gravitational parameter, 2ρgap3 /9µD∞ dimensionless gap width, s/ap unit vector particles flux vector, m−2 s−1 particles flux in r-direction, m−2 s−1 particles flux normal to the collector surface, m−2 s−1 Boltzmann constant, J/K particle deposition rate constant, m/s favorable and unfavorable deposition rate constants, m/s characteristic length, m particle buoyant mass, kg local number concentration, m−3 number of collectors per unit volume, m−3 number concentration of particles at bed inlet, m−3 number concentration of particles far from a collector, m−3 attraction number, AH ac2 /9π µap4 U∞ Asph gravity number, 2ρgac2 /9µU∞ Asph electrostatic (repulsion) number, 6π ǫζc ζp ap /AH Peclet number, defined differently for the various geometries source term, m−3 s−1 radial coordinate, m Reynolds number (see Table 13.3)

530

DEPOSITION OF COLLOIDAL PARTICLES

s S Sc Sh Sh(θ), Shl St t T ur uz u u∗ ust ush u∞ vr vt vz v vm x x X

dimensional gap between two surfaces, s = ap h collector surface area, m2 Schmidt number, (µ/ρ)/D∞ average Sherwood number (see Table 13.3) local Sherwood number Stokes number, Eq. (13.23) time, s absolute temperature, K radial fluid velocity, m/s normal fluid velocity, m/s fluid velocity vector, m/s dimensionless fluid velocity vector stagnation fluid velocity vector, m/s shear fluid velocity vector, m/s characteristic velocity; fluid velocity far from a collector; superficial velocity in a packed bed, m/s radial component of particle velocity, m/s terminal velocity of a free settling spherical particle, m/s normal component of particle velocity, m/s particle velocity vector, m/s average velocity in a channel, m/s Cartesian coordinate, distance along a channel, m direction vector, m dimensionless direction vector, Eq. (13.21)

Greek Symbols α αc − α ρ ǫ ζc ζp η ηI θ θ, θf , θu κ λ λ¯ µ ν ρ

impinging jet flow coefficient, m−1 s−1 volume fraction of collector particles dimensionless impinging jet flow coefficient density difference between particle and suspending fluid, kg/m3 permittivity of liquid, C/Vm collector zeta potential, V colloidal particle zeta potential, V collection efficiency pure interception collection efficiency angular coordinate fractional surface coverage inverse Debye length, m−1 filter coefficient, Eq. (13.15), m−1 London wavelength, of order of 10−7 m dimensionless London wavelength, λ/ap fluid viscosity, Pa s kinematic viscosity, m2 /s fluid density, kg/m3

13.10

PROBLEMS

531

particle density, kg/m3 stream function, m3 /s for flow over a spherical particle; m2 /s for flow over a cylinder limiting stream function defining a particular collection mode angular velocity of a spinning disc, radians/s local radial coordinate, m del operator, m−1 Laplacian operator, m−2

ρp ψ ψL ω ̟ ∇ ∇2 13.10

PROBLEMS

13.1. A spherical particle of radius a contained in a large vessel is released from a gap height of h0 . The vessel contains a liquid of viscosity µ. The density difference between heavy particle and the liquid is ρ. Gravity acts downwards. The geometry for the approach of the particle to the bottom surface of the vessel, which is assumed planar, is shown in Figure 13.37. (a) Assume that gravity is the sole force acting on the particle. Show that under pseudo steady-state conditions 2a 2 ρg dh =− f1 dt 9µ where g is the gravitational constant and f1 is the universal hydrodynamic correction function that accounts for the presence of the vessel bottom surface. (b) Let the van der Waals force be included in the force balance. Show that  2  dh 2a ρg AH =− + f1 dt 9µ 36π µh2

Figure 13.37. Schematic depiction of a particle approaching a planar surface of a vessel.

532

DEPOSITION OF COLLOIDAL PARTICLES

where AH is the Hamaker constant and the van der Waals interaction energy between a sphere and a flat surface is given by −AH a/(6h), see Table 11.2. (c) Let the dimensionless time, τ , be defined as τ=

2ρga t 9µ

and dimensionless length, H , as H =

h a

Show that

where

  G dH = − 1 + 2 f1 dτ H G=

AH 8π a 4 ρg

Here, G is a dimensionless group that accounts for the ratio of attractive to gravitational forces. (d) The hydrodynamic correction function, for convenience, can be taken as f1 =

H H +1

Show that τ =−



H

H0

H (H + 1) dH H2 + G

where H0 = h0 /a at time zero. (e) Let a = 10−5 m, ρ = 1000 kg/m3 , µ = 10−3 Pa.s, and H0 = 5. Consider four values of the Hamaker constant, namely, AH = 0, 10−21 , 10−20 , and 10−19 J. For each AH value, plot the variation of the scaled separation H with τ . Comment on your solution. What is your observation for G = 0? 13.2. (a) Derive the equivalent expression for dH /dτ in the presence of gravitational, van der Waals, and electrostatic repulsion forces in a manner similar to part (c) of Problem 13.1. Assume the zeta potential of the particle is the same as the surface of the vessel. What is the additional dimensionless group and how would you characterize it? (b) Plot the variation of the dimensionless separation gap H with time τ for zeta potentials of 50 mV on the sphere and the vessel surfaces for water at 20◦ C containing 0.001 M NaCl. Assume AH = 10−20 J. Compare with the case of zero repulsive force. All other physical data are the same as in Problem 13.1.

13.11

13.11

REFERENCES

533

REFERENCES

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Bowen, B. D., Levine, S., and Epstein, N., Fine particle deposition in laminar flow through parallel-plate and cylindrical channels, J. Colloid Interface Sci., 55, 275–290, (1976). Brenner, H., The slow motion of a sphere through a viscous fluid towards a plane surface, Chem. Eng. Sci., 16, 242–251, (1961). Chen, J. Y., Ko, C.-H., Bhattacharjee, S., and Elimelech, M., Role of spatial distribution of porous medium surface charge heterogeneity in colloid transport, Colloids Surf. A, 191, 3–15, (2001). Clift, R., Grace, J. R., and Weber, M. E., Bubbles, Drops and Particles, Academic Press, London, (1978). Clint, G. E., Clint, J. G., Corkill, J. M., and Walker, T., Deposition of latex particle on a planar surface, J. Colloid Interface Sci., 44, 121–132, (1973). Dabros, T., and van de Ven, T. G. M., A direct method for studying particle deposition onto solid surfaces, Colloid Polym. Sci., 261, 694–707, (1983). Dabros, T., and van de Ven, T. G. M., Deposition of latex particles on glass surfaces in an impinging jet, Physicochem. Hydrodyn., 8, 161–172, (1987). Elimelech, M., Gregory, J., Zia, X., and Williams, R. A., Particle Deposition and Aggregation: Measurement, Modelling, and Simulation, Butterworth, London, (1995). Feder, J., Random sequential adsorption, J. Theor. Biol., 87, 237–254, (1980). Fitzpatrick, J. A., Mechanisms for particle capture in water filtration, PhD Dissertation, Division of Engineering and Applied Physics, Harvard University, Cambridge, MA, (1972). Fitzpatrick, J. A., and Spielman, L. A., Filtration of aqueous latex suspensions through beds of glass spheres, J. Colloids Interface Sci., 43, 350–369, (1973). Friedlander, S. K., Theory of aerosol filtration, Ind. Eng. Chem., 50, 1161–1164, (1958). Friedlander, S. K., Smoke, Dust and Haze: Fundamentals of Aerosol Dynamics, 2nd ed., Oxford University Press, New York, (2000). Fuchs, N. A., The Mechanics of Aerosols, Dover, New York, (1964). Goldman, A. J., Cox, R. G., and Brenner, H., Slow viscous motion of a sphere parallel to a plane wall: I. Motion through a quiescent fluid, Chem. Eng. Sci., 22, 637–651, (1967a). Goldman, A. J., Cox, R. G., and Brenner, M., Slow viscous motion of a sphere parallel to a plane wall: II. Couette flow, Chem. Eng. Sci., 22, 653–660, (1967b). Goren, S. L., The normal force exerted by creeping flow on a small sphere touching a plane, J. Fluid Mech., 41, 619–625, (1970). Goren, S. L., and O’Neill, M. E., On the hydrodynamic resistance to a particle of a dilute suspension when in the neighborhood of a large obstacle, Chem. Eng. Sci., 26, 325–338, (1971). Griffin, F. O., and Meisen, A., Impaction of spherical particles on cylinders at moderate Reynolds numbers, Chem. Eng. Sci., 28, 2155–2164, (1973). Guzy, C. J., Bonano, E. J., and Davis, E. J., The analysis of flow and colloidal particle retention in fibrous porous media, J. Colloid Interface Sci., 95, 523–543, (1983). Happel, J., Viscous flow in multiparticle systems, AIChE J., 4, 197–201, (1958). Happel, J., Viscous flow relative to arrays of of cylinders, AIChE J., 5, 174–177, (1959). Happel, J., and Brenner, H., Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, (1965).

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Hirtzel, C. S., and Rajagopalan, R., Colloidal Phenomena, Advanced Topics, Noyes, NJ, (1985). Hogg, R., Healy, T. W., and Fuerstenau, D. E., Mutual coagulation of colloidal dispersions, Trans. Faraday Soc., 63, 1638–1651, (1966). Honig, E. P., Roeberson, G. J., and Wiersema, P. H., Effect of hydrodynamic interaction on the coagulation rate of hydrophobic colloids, J. Colloid Interface Sci., 36, 97–109, (1971). Hull, M., and Kitchener, J. A., Interaction of spherical colloidal particles with planar surfaces, Trans. Faraday Soc., 65, 3093–3104, (1969). Israelachvili, J. N., Intermolecular and Surface Forces, 2nd ed., Academic Press, London, (1992). Israelachvili, J. N., and Pashley, R. M., Measurement of the hydrophobic interaction between two hydrophobic surfaces in aqueous electrolyte solutions, J. Colloid Interface Sci., 98, 500–514, (1984). Jeffrey, D. J., and Onishi, Y., Calculations of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds number flow, J. Fluid Mech., 139, 261–290, (1984). Johnson, P. R., Sun, N., and Elimelech, M., Colloid transport in geochemically heterogeneous porous media: Modeling and measurements, Environ. Sci. Technol., 30, 3284–3293, (1996). Kihira, H., Ryde, N., and Matijevic, E., Kinetics of heterocoagulation. Part 2: The effect of discreteness of surface charge, J. Chem. Soc. Faraday Trans., 88, 2379–2386, (1992). Ko, C.-H., Bhattacharjee, S., and Elimelech, M., Coupled model of colloidal and hydrodynamic interactions on the RSA dynamic blocking function for particle deposition onto packed spherical collectors, J. Colloid Interface Sci., 229, 554–567, (2000). Levesque, M. , Ann. Mines, 13, 201, (1928). Levich, V. G., Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, (1962). Liu, J., Xu, Z., and Masliyah, J. H., Role of fine clays in bitumen extraction from oil sands, AIChE J., 50, 1917–1927, (2004). Loveland, J. P., Bhattacharjee, S., Ryan, J. N., and Elimelech, M., Colloid transport in a geochemically heterogeneous porous medium: Aquifer tank experiment and modeling, J. Contaminant Hydrology, 65, 161–182, (2003). Marshall, J. K., and Kitchener, J. A., The deposition of colloidal particles on smooth solids, J. Colloid Interface Sci., 22, 342–351, (1966). Masliyah, J. H., and Duff, A., Impingement of spherical particles on elliptical cylinder, Aerosol Sci., 6, 31–43, (1975). Masliyah, J. H., and Epstein, N., Numerical solution of heat and mass transfer from spheroids in steady axisymmetric flow, Prog. Heat Mass Transf., 6, 613–632, (1973). Michael, D. M., and Norey, P. W., Particle collision efficiencies for a sphere, J. Fluid Mech., 37, 567–575, (1969). Neale, G., and Masliyah, J. H., Flow perpendicular to mats of randomly arranged cylindrical fibers (Importance of cell models), AIChE J., 21, 805–807, (1975). Prieve, D. C., and Lin, M. M. J., Adsorption of Brownian hydrosols onto rotating disc aided by uniform applied force, J. Colloid Interface Sci., 76, 32–47, (1980). Prieve, D. C., and Ruckenstein, E., Effect of London forces upon the rate of deposition of Brownian particles, AIChE J, 20, 1178–1187, (1974). Probstein, R. F., Physicochemical Hydrodynamics, An Introduction, 2nd ed., Wiley Interscience, New York, (2003).

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Rabinovich, Y. I., and Yoon, R.-H., Use of atomic force microscope for the measurements of hydrophobic forces between silanated silica plate and glass sphere, Langmuir, 10, 1903– 1909, (1994). Rajagopalan, R., and Kim, J. S., Adsorption of Brownian Particles in the presence of potential barriers: Effect of different modes of double-layer interaction, J. Colloid Interface Sci., 83, 428–448, (1981). Rajagopalan, R., and Tien, C., Trajectory analysis of deep bed filtration with sphere-in-cell porous media model, AIChE J., 22, 523–533, (1976). Sanders, R. S., Chow, R. S., and Masliyah, J. H., Deposition of bitumen and asphaltene stabilized emulsions in an impinging jet cell, J. Colloid Interface Sci., 174, 230–245, (1995). Sanders, R. S., Chow, R. S., and Masliyah, J. H., Hydrophobic interactions in silane treated silica suspensions and bitumen emulsions, Can. J. Chem. Eng., 81, 43–52, (2003). Schaaf, P., and Talbot, J., Kinetics of random sequential adsorption, Phys. Rev. Lett., 62, 175–178, (1989). Song, S., Lopez-Valdivieso, A., Reyes-Bahena, J. L., Bermejo-Perez, H. I., and Trass, O., Hydrophobic flocculation of galena fines in aqueous suspensions, J. Colloid Interface Sci., 227, 272–281, (2000). Spielman, L. A., Particle capture from low-speed laminar flows, Ann. Rev. Fluid Mech., 9, 297–319, (1977). Spielman, L. A., and Cukor, P. M., Deposition of non-Brownian particles under colloidal forces, J. Colloid Interface Sci., 43, 51–65, (1973). Spielman, L. A., and Fitzpatrick, J. A., Theory for particle collection under London and gravity forces, J. Colloid Interface Sci., 42, 607–623, (1973). Spielman, L. A., and Goren, S. L., Capture of small particles by london forces from low-speed liquid flows, Environ. Sci. and Tech., 4, 135–140 (see corrections in 5, 254, 1971), (1970). Suzuki, A., Ho, N. F. H., and Higuchi, W. I., Predictions of the particle size distribution changes in emulsions and suspensions by digital computation, J. Colloid Interface Sci., 29, 552–564, (1969). Tien, C., Granular Filtration of Aerosols and Hydrosols, Butterworths, Boston, (1989). Usui, S., Interaction of electrical double layers at constant surface charge, J. Colloid Interface Sci., 44, 107, (1973). van de Ven, T. G. M., Colloidal Hydrodynamics, Academic Press, London, (1989). Vaidyanathan, R., and Tien, C., Hydrosol deposition in granular beds, Chem. Eng. Sci., 43, 289–302, (1988). Weber, M. E., and Paddock, D., Interceptional and gravitational collision efficiencies for single collectors at intermediate Reynolds numbers, J. Colloid Interface Sci., 94, 328–335, (1983). Xu, Z., and Yoon, R.-H., The role of hydrophobic interactions in coagulation, J. Colloid Interface Sci., 132, 532–541, (1989). Xu, Z., and Yoon, R.-H., A study of hydrophobic coagulation, J. Colloid Interface Sci., 134, 427–434, (1990). Yao, K. M., Habibian, M. T., and O’Melia, C. R., Water and wastewater filtration: Concepts and applications, Environ. Sci. Technol., 5, 1105–1112, (1971).

CHAPTER 14

NUMERICAL SIMULATION OF ELECTROKINETIC PHENOMENA

Application of numerical solution techniques often becomes the only recourse when one intends to employ the electrokinetic models discussed in the earlier chapters to complex systems. It was noted in the earlier chapters, that the closed form mathematical expressions describing various electrokinetic phenomena were derived for low surface potentials and highly simplified geometries using linear perturbation approximations. Such approximations and simplifications may potentially restrict the application of analytical results when addressing more complicated systems of practical interest. For instance, the electric potential and free charge distribution used in the treatment of electroosmosis and electrophoresis were obtained from a solution of the linearized Poisson–Boltzmann equation, which is strictly valid for low surface potentials, namely |ψs | < 25 mV. For higher surface potentials, one must solve the full non-linear Poisson–Boltzmann equation. This becomes a difficult task if one attempts analytic routes, particularly for complex geometries. Furthermore, analytic solutions of the non-linear Poisson–Boltzmann equation are only feasible for symmetric electrolytes. Modeling electrokinetic transport phenomena requires simultaneous solution of at least three coupled physical models, namely, the governing equations for electrostatics (Poisson equation), fluid flow (Navier–Stokes and continuity equations), and transport of ions (Nernst–Planck equations). For a multicomponent system in a threedimensional flow, this translates into solving n coupled Nernst–Planck equations for the n ionic species, the Poisson equation, three equations for momentum conservation along the three coordinate directions,and the fluid continuity equation. This Electrokinetic and Colloid Transport Phenomena, by Jacob H. Masliyah and Subir Bhattacharjee Copyright © 2006 John Wiley & Sons, Inc.

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implies that, in general n + 5 partial differential equations must be solved simultaneously for an electrokinetic problem involving n ionic species. The equations are coupled as follows: The Poisson equation relates the electric potential distribution to the volumetric charge density, the latter being related to the distribution of the ions. The ion distributions are provided by the Nernst–Planck equations. The Nernst–Planck equation requires the fluid velocity components to obtain the convective flux, as well as the electric field to obtain the migration flux. The velocity components are provided by the Navier–Stokes (and continuity) equations, while the electric field is provided by the solution of the Poisson equation. Finally, the solution of the Navier–Stokes equations involves consideration of the electrical body force, which is determined from the Poisson and the Nernst–Planck equations. Even for relatively simple geometries, the task of solving all these equations simultaneously is quite formidable. In this chapter, we will discuss the key issues involved in attempting numerical solutions for electrokinetic problems. The objective of this chapter is not to expound on basic numerical techniques. One should refer to appropriate textbooks on numerical methods for that purpose. Here, we specifically illustrate, through some examples, how one can implement a numerical scheme to solve typical electrokinetic transport problems, which are cast as a set of coupled partial differential equations. Numerous sophisticated techniques exist for solution of partial differential equations numerically. In this chapter, we use a fairly generic approach, namely, finite element analysis, for the solution of the governing electrokinetic equations. This, of course, does not imply that finite element technique is the only approach to obtain a viable solution of electrokinetic problems. Nearly any technique for solving coupled partial differential equations, like finite difference, finite volume, boundary element, method of lines, and spectral techniques, to name a few, can be used to solve such problems numerically. The sole advantage of the finite element technique seems to be its ability to provide a solution for arbitrary geometries and computational domains. Furthermore, since numerous commercial software implementing the finite element technique are readily available, it was felt that relegating the numerical task to a robust software will allow one to focus on the problem formulation, which is our key objective in this chapter.

14.1 TOOLS AND METHODS FOR COMPUTER BASED SIMULATIONS The models for electrokinetic transport are continuum models, which yield partial differential equations. Combined, the Poisson, the Navier–Stokes, and the Nernst–Planck equations span the entire spectrum of generic classification of partial differential equations (PDE), representing parabolic, elliptic, and hyperbolic equations. The Poisson equation is an example of elliptic PDEs. The Nernst–Planck equations are generally parabolic, but depending on parameter values, may sometimes behave as hyperbolic equations. The Navier–Stokes equations are elliptic for quasi-steady creeping flow regimes (no inertia term), but the general Navier–Stokes equation can assume all

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three forms of PDEs depending on parameter values. An essential component for attempting to solve these equations simultaneously is to have a numerical technique for solving PDEs that is generic enough to solve elliptic, parabolic, and hyperbolic equations. Furthermore, most of these equations are non-linear, which necessitates a robust equation solver that can handle non-linearities. Adding to this list, the requirement of handling arbitrary geometries and computational domains leaves very few options other than to use techniques like finite difference and finite element analysis. In this chapter, we will use the finite element technique owing to the fact that most finite element solvers are quite generic in nature to accommodate diverse types of equations, and allow handling of different geometries quite readily. Several modeling software based on finite element technique have become available over the past decade, including ANSYS, Coventor, FEMLab, Fluent, ALGOR, etc. These software generally allow (to varying degrees of sophistication) what is commonly referred to as multiphysics modeling, and generally provide tools and algorithms to deal with electrokinetic problems associated with microfluidic simulations. The computational results presented in this chapter were obtained using one such commercially available software, FEMLab. Relegating the numerical aspects of the finite element implementation to a commercial software allows us to focus on the specific issues related to formulation of the electrokinetic problems to render them amenable to numerical solution. This brings us to an important fact one should keep in mind when using numerical techniques. The key to successfully implementing a numerical methodology is to realize at the outset that numerical methods are approximate, and generally work within a range of real numbers roughly spanning eight orders of magnitude (in single precision). As an example, if one is to solve a partial differential equation with a dependent variable that ranges from 1 to 0, the numerical solution will only provide reasonably accurate predictions of the variable over a range spanning 1 to 10−8 . If instead, the dependent variable ranges from 105 to 0, perhaps the smallest value that will be accurately predicted by the numerical solution is 10−3 . The problem is exacerbated when we deal with coupled partial differential equations. In this case, one needs to carefully scale all the dependent (and independent) variables such that they are of comparable magnitudes. In this context, the most important tools for implementation of a numerical solution are still pen and paper. One should perhaps spend ample time in properly setting up the model problem such that it can be solved readily using a standard numerical technique. This can be achieved by careful non-dimensionalization and scaling of the governing equations, appropriate simplification of the geometry, careful choice of the parameter phase space, use of symmetry, coordinate transformation, and a variety of other techniques. None of these initial steps require a computer or any software. Nevertheless, spending time on these aspects is important if one desires to obtain a quick and sufficiently accurate numerical solution for any mathematical model. There are several challenging issues specific to electrokinetic problems that need consideration during numerical simulations. The foremost of these is the frequent use of boundary conditions at infinity when defining an electrokinetic problem. Consider for instance, solution of the Poisson–Boltzmann equation for the electric potential

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distribution in an electrolyte near an infinite planar charged surface (see Section 5.2). The boundary conditions used for this problem are: 1. at the surface of the plate, the electric potential is equal to the surface potential of the plate, and, 2. at an infinite distance away from the plate, the surface potential and the electric field in the direction normal to the plate are zero. The second boundary condition at infinity poses a serious problem in numerical computations. One cannot readily implement infinity in a numerical solution technique. Consequently, one must implement an alternative approach to define the boundary conditions in a numerical problem. A coordinate transformation is the commonly adopted approach, whereby the infinity is somehow converted to a finite number. This type of transformation will render such problems amenable to a numerical treatment. It is also possible to artificially implement the outer boundary (corresponding to infinity) at a sufficiently large distance away from the plate, such that the electric field and potential can be assumed to be infinitesimally small at this location. This type of artificial imposition of a boundary condition at infinity cannot be considered a sound mathematical practice, since one cannot a priori tell whether a given distance is sufficiently far away to actually validate the assumption of infinitesimal potential or field at this location. Therefore, whenever such artificial boundary conditions are used, the validity of the methodology should be verified by placing the boundary further and further away and comparing the solutions obtained at these different locations. A second issue arising in numerical solution of electrokinetic problems stems from the high non-linearity of the Poisson equation. Consider the Poisson–Boltzmann equation, for example. We observed in the earlier chapters that for high electrolyte concentrations, when the Debye screening length, κ −1 , is small, the analytical solution of the Poisson–Boltzmann equation is quite straightforward. This is not the case for numerical solutions. At high electrolyte concentrations, due to small Debye screening lengths, the electrical potential decays so rapidly near the charged surface, that many standard numerical differential equation solvers fail to capture the sharply decaying profile of the electric potential. This gives rise to convergence problems, or results that are grossly erroneous. Contrary to what was observed in case of most analytic solutions of electrostatic double layer interactions, numerical approaches work quite well when we consider the solution of the Poisson–Boltzmann equation at lower electrolyte concentrations or larger values of κ −1 . There are several other issues of the above nature that one needs to consider during a numerical solution of an electrokinetic problem. We will discuss some of these in the following sections. In this chapter, the application of numerical techniques will be discussed in context of three problems. First, a solution of the Poisson–Boltzmann equation will be obtained in a cylindrical capillary, which contains two colloidal particles with their centers lying on the capillary axis. This is a highly non-linear problem, which involves solution of an elliptic partial differential equation for a single dependent variable, namely, the electric potential. In context of this problem, several issues regarding numerical implementation of finite element approximation will be discussed. The

14.2

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541

second problem to be discussed involves electrokinetic flow of an electrolyte solution in a charged cylindrical capillary under the combined influence of pressure and electric potential gradient. This will serve as an example of typical multiphysics problems associated with electrokinetic transport phenomena. We will solve both steady-state and time dependent versions of the governing equations.A specific case of the problem involving solely pressure driven flow will provide a direct measure of the streaming potential developed across the capillary. Finally, the third problem we consider in this chapter involves simulation of the electrophoretic mobility of a spherical particle under the influence of an externally imposed electric potential gradient. All three problems will be solved using the general forms of the governing equations obtained from the continuum models of electrostatics and electrokinetics. These problems are discussed with the goal of comparing the numerical results with various analytical results derived in the previous chapters. The comparisons will, on one hand, provide a measure of the accuracy of the numerical calculations. On the other hand, the comparisons will show that although approximate, some of the analytical results derived in the previous chapters are remarkably accurate.

14.2 NUMERICAL SOLUTION OF THE POISSON–BOLTZMANN EQUATION The electric double layer (EDL) interaction energy or force between two colloidal particles is of critical importance in prediction of colloidal stability as noted in Chapter 11. The EDL force is susceptible to tremendous variations with physico-chemical conditions like the electrolyte concentration, the charging behavior of the particles (constant potential or constant charge), the dielectric constant of the solvent, as well as the shape of the particles. In many problems, for instance, particle transport in porous media, membrane filtration, chromatographic separations, and capillary electrophoresis, the colloidal particles are usually confined in narrow electrolyte filled capillaries. For modeling purposes, the capillaries are often assumed to be straight circular cylinders. Evaluating the EDL interaction force on a single colloidal particle trapped inside a narrow capillary is a complex task. An analytic solution of the problem exists for low potentials on the capillary wall and particle surfaces, which allows linearization of the governing Poisson–Boltzmann equation (Smith III and Deen, 1980). This solution is valid for symmetric electrolytes. Analytical solution for this problem does not exist for higher potentials on the interacting surfaces of the capillary and the particle, where the complete (non-linear) Poisson–Boltzmann equation needs to be solved. The problem becomes far more complex when we consider two particles trapped inside the capillary. There is no analytic solution of the Poisson– Boltzmann equation, even for a symmetric electrolyte, for this problem. Numerical solutions for this problem, however, have been reported (Bowen and Sharif, 1998; Gray et al., 1999). The calculation of the EDL interaction force between two spherical colloidal particles lying on the axis of a straight cylindrical capillary, with their surfaces separated by a distance h, forms an interesting case study in numerical analysis. The geometry

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Figure 14.1. Schematic representation of two spherical colloidal particles interacting within a straight cylindrical capillary. The particles lie on the axis of the cylinder, and are separated by a distance h.

is schematically depicted in Figure 14.1. Here, two charged particles of radius a with surface potential (zeta potential) ψp are suspended in an electrolyte solution confined within an infinitely long capillary of radius b. The capillary wall has a surface potential of ψc . This problem represents a class of many-body (three-body to be specific) electrostatic interactions, where the interaction energy between the two spheres is influenced by the presence of the charged capillary wall. The first two numerical studies of this problem (Bowen and Sharif, 1998, 1999; Gray et al., 1999) appeared in the late nineties, both employing the finite element technique. Among these, the numerical results of Bowen and Sharif (1998) incorrectly showed that there will be an attractive force between two similarly charged spherical particles at large separation distances when the particles are trapped inside a cylindrical capillary. Their study was spurred by the experiments of Larsen and Grier (1997), which indicated the presence of a long-range attraction between two similarly charged colloidal particles. The numerical predictions of Gray et al. (1999), however, did not show the presence of any attractive force between the spherical particles, although they employed the exact problem formulation used by Bowen and Sharif (1998). Later, it was theoretically demonstrated that the long-range attraction observed in the experiments of Larsen and Grier (1997) can not be replicated in the framework of the Poisson–Boltzmann equation (Sader and Chan, 1999; Sader and Chan, 2002). We choose this problem as our model to demonstrate the application of finite element techniques for solution of the Poisson–Boltzmann equation. Apart from revisiting a topic of considerable recent interest, and addressing a geometrical system that is relevant to electrokinetic transport phenomena, the model also allows numerical evaluation of the interaction force between two spherical particles as a limiting case. This limiting case is obtained when the capillary radius, b, is considerably larger than the radius of the spheres, a. A variety of numerical solution schemes for the Poisson– Boltzmann equation to evaluate the interaction force between two spheres immersed in an infinitely large electrolyte bath are available (Glendinning and Russel, 1983; Carnie et al., 1994; Stankovich and Carnie, 1996; Warszynski and Adamczyk, 1997). Amongst these, the solution of Glendinning and Russel (1983) is based on a multipole expansion technique, and applies to the linearized Poisson–Boltzmann equation for

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543

two equal spheres in a symmetric electrolyte. The remaining studies are based on a finite difference approach, employing a bispherical coordinate system to tackle the boundary condition at infinity. Use of finite element and boundary element techniques for solving the problem have also been reported (Bowen and Sharif, 1997; Grant and Saville, 1995). Incidentally, the boundary element technique can only be applied to solve the linearized Poisson–Boltzmann equation. For the geometry of the present problem, the cylindrical coordinate system (r, θ, x) appears to be the most convenient, since it allows the utilization of axial symmetry (invariance of the parameters with respect to the angular coordinate, θ) to render the three dimensional geometry using a two dimensional model. The typical steps of a numerical solution procedure based on finite element analysis involve: (i) model formulation, (ii) design of the computational geometry, (iii) mesh generation, (iv) solution of the governing equations, (v) postprocessing, and (vi) model validation. In the following, these steps are described, focussing on how the approach is employed for the solution of the Poisson–Boltzmann equation.

14.2.1 Problem Formulation For a symmetric (z : z) electrolyte, the Poisson–Boltzmann equation can be written as ∇ 2  = κ 2 sinh()

(14.1)

where  (= zeψ/kB T ) is the scaled electric potential, z is the ionic valence, e is the magnitude of the electronic charge, ψ is the electrical potential (V). The parameter κ in Eq. (14.1) is the inverse Debye length, given by

κ=




2z2 e2 n∞ ǫkB T

(14.2)

where n∞ is the bulk electrolyte concentration (m−3 ) and ǫ is the dielectric permittivity of the electrolyte solution (C2 /Nm2 ). The Poisson–Boltzmann equation (14.1) is applied to the electrolyte filled region in the capillary (Figure 14.1). When the particle centers reside on the axis of the cylindrical capillary, one can employ axial symmetry to represent the problem in a two-dimensional axisymmetric cylindrical coordinate system (r, x) as shown in Figure 14.2. The explicit form of Eq. (14.1) in axisymmetric cylindrical coordinates is ∂ 2 1 ∂ ∂ 2 + + = κ 2 sinh() ∂r 2 r ∂r ∂x 2

(14.3)

Note that use of axial symmetry to render the problem two dimensional is only possible when the particle centers are on the axis of the capillary. If, further, one scales the coordinates r and x with respect to the Debye screening length, κ −1 , such that r¯ = κr

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Figure 14.2. Schematic representation of the geometrical domain used in the finite element analysis. The point O, located at the center of the sphere BCD, is the origin of the cylindrical coordinate system.

and x¯ = κx,1 Eq. (14.3) becomes ∂ 2 1 ∂ ∂ 2 + + = sinh() 2 ∂ r¯ r¯ ∂ r¯ ∂ x¯ 2

(14.4)

Equation (14.4) is the non-dimensional form of the Poisson–Boltzmann equation written in axisymmetric cylindrical coordinates. We now define the boundary conditions for this equation. Referring to Figure 14.2, and assuming constant surface potentials on the spheres and the cylindrical capillary, we can write  = p

for ∂ ∈ BCD and EFG

(14.5-a)

for ∂ ∈ IJ

(14.5-b)

 = c

The above conditions specify the surface potentials on the spheres and the capillary wall, respectively. Here, the potential on both spheres are assumed to be equal (p ), while the potential on the capillary wall, c , may be different from the particle surface potentials. On the axis of the cylindrical capillary, namely, segments AB, DE, and GH (Figure 14.2), one can employ the symmetry condition, stated as ¯ =0 n · ∇

for ∂ ∈ AB, DE, and GH

(14.5-c)

In the above equation, n represents a unit normal to the surface. In the present system, the axial symmetry simply means that at r = 0, ∂/∂ r¯ = 0. We are now left only with two line segments, namely, AJ and HI, where the boundary conditions need to be defined. Unfortunately, there are no well-defined boundary 1

In this case, the gradient operator, ∇, can be scaled as   ∂ ∂ ir¯ + ix¯ = κ −1 ∇ ∇¯ = ∂ r¯ ∂ x¯

¯ respectively. where ir¯ and ix¯ represent unit vectors along the scaled coordinates, r¯ and x,

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545

conditions for these two segments unless these are located at infinity. In a numerical solution, these lines cannot be assumed to be located at infinite distances from the origin. Let us first consider what boundary conditions can be applied on segments AJ and HI, if there are no particles in the capillary. In this case, if the capillary is assumed to be infinitely long, then the electric field component along the axial direction will be zero on AJ and HI. In other words, ∂/∂ x¯ = 0 on these two segments. Thus, one can assume that if AJ and HI are sufficiently far away from the particles, the electric field at these segments will have negligible influence of the particles. In other words, placing AJ and HI sufficiently far away from the particles allows the application of a symmetry boundary condition of the form ¯ =0 n · ∇

for ∂ ∈ AJ and HI

(14.5-d)

This condition is not exactly a true boundary condition. The validity of this artificial and approximate boundary condition must be verified once the numerical solution is obtained. However, it is discernable that as AJ and HI are moved further away from the particles, the accuracy of this boundary condition will improve. Equations (14.5-a)–(14.5-d) represent the boundary conditions for the Poisson– Boltzmann equation applied to the geometry of Figure 14.2. If, instead of constant surface potential, constant charge density conditions are applied on the particles and the capillary wall, Eqs. (14.5-a) and (14.5-b) are replaced by constant surface charge boundary conditions, given by ¯ = −n · ∇

ze qp = σp κǫkB T

and ¯ = −n · ∇

for ∂ ∈ BCD and EFG

ze qc = σc κǫkB T

for ∂ ∈ IJ

(14.6-a)

(14.6-b)

Here, qp and qc are the charge densities (C/m2 ) on the surfaces of the particles and the capillary wall, respectively, while σp and σc are the corresponding non-dimensional surface charge densities. Equation (14.4), along with Eqs. (14.5-a)–(14.5-d) can be solved using an appropriate numerical technique. This will provide the results for constant potential surfaces. Replacing Eqs. (14.5-a) and (14.5-b) with (14.6-a) and (14.6-b), will yield the corresponding results for constant surface charge boundary conditions. The above formulation can be considered as fairly general, and should provide insight into the behavior of EDL interactions for a wide range of conditions. One should note a few simplifications made in the formulation of the problem. First, utilization of axial symmetry reduces the dimensionality of the problem (from a three dimensional domain to an axisymmetric two dimensional plane). Secondly, scaling the length with respect to the screening length of the EDL reduces the number of independent variables – one can now simply perform the calculations for different selected values of the parameter κa, where a is the particle radius, without explicitly having to consider κ −1 and a separately. Furthermore, such scaling makes the implementation of the numerical

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NUMERICAL SIMULATION OF ELECTROKINETIC PHENOMENA

procedure more facile, since one does not need to deal with real length scales of the system anymore. 14.2.2 Finite Element Formulation The finite element formulation employs the so-called general form of a partial differential equation, written in rectangular cartesian coordinates (x, y, z), given by ∂φ +∇ ·Ŵ =R ∂t

(14.7)

where φ is the dependent variable, t is time, ∇ is the gradient operator in cartesian coordinates, Ŵ represents the spatial gradient of the dependent variable, and R is the source term or a forcing function. For a time-independent problem, Eq. (14.7) becomes ∇ ·Ŵ =R (14.8) The function Ŵ is written in a two-dimensional cartesian coordinate system (x, y) as Ŵ = ∇φ =

∂φ ∂φ ix + iy ∂x ∂y

(14.9)

where ix and iy are unit vectors along the x and y coordinate directions, respectively, and the gradient operator in the (x, y) coordinate system is ∇=

∂ ∂ ix + iy ∂x ∂y

(14.10)

Using Eqs. (14.9) and (14.10), Eq. (14.8) can be written as 

   ∂ ∂ ∂φ ∂ 2φ ∂φ ∂ 2φ ix + iy · ix + iy = + =R ∂x ∂y ∂x ∂y ∂x 2 ∂y 2

(14.11)

The forcing function, R, can be any function of φ, ∇φ, x, and y, and is obtained from the governing physics of the problem. Typically, most finite element codes are developed to solve a partial differential equation in the general form written in cartesian coordinates represented by Eq. (14.7) or Eq. (14.8). Employing such a cartesian computational domain simplifies conversion of the partial differential equations from the general form to the so-called “weak form,” which is often a necessary step in implementation of finite element techniques (Zienkiewicz and Taylor, 1989). Consequently, when one does not develop the entire finite element code for solving a problem in a specific coordinate system, but relies on either standard software, or uses a “black-box” code, it is perhaps safer to recast any given model into the general form and solve the problem in a cartesian-type coordinate frame. We now investigate how this can be achieved for our model problem, which was developed in an axisymmetric cylindrical coordinate system.

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547

Equation (14.4), which represents the Poisson–Boltzmann equation in a cylindrical coordinate system, can be rearranged as ∂ 2 ∂ 2 1 ∂ + = sinh() − 2 ∂ r¯ ∂ x¯ 2 r¯ ∂ r¯

(14.12)

It is immediately evident from Eqs. (14.11) and (14.12), that the left hand side of Eq. (14.12) can be treated as the Laplacian operator written in cartesian coordinates. In this case, one obtains a modified form of the forcing function, given by R = sinh() −

1 ∂ r¯ ∂ r¯

(14.13)

The form of Eq. (14.13) implies that the term (∂/∂ r¯ )/¯r arising from the Laplacian operator in the cylindrical coordinate system can be added to the forcing function term, R. This rearrangement renders the resulting partial differential equation to behave like one written in a cartesian coordinate system. Note that the solution of Eq. (14.12) will still provide the variation of the dependent variable  with the scaled radial (¯r ) and axial (x) ¯ coordinates in the cylindrical coordinate system. However, the numerical code will “see” the governing equation as one written in a cartesian coordinate system, with a slightly different forcing function. This technique allows a computer code written for solving a partial differential equation in the cartesian coordinate system the flexibility of tackling other types of coordinate systems, as long as it can handle different definitions of the forcing function R. Writing the non-dimensional Poisson–Boltzmann equation in the form of Eq. (14.12) allows its solution in many standard finite element programs. The finite element method approximates the spatial variation of the dependent variable (φ) as a combination of piecewise polynomial shape functions. The shape function is defined such that it has a value of 1 at a given node in an element and smoothly approaches a value of zero at all other nodes of the element. The sum of these shape functions, multiplied with appropriate weights, over all the nodes constituting the element provides the approximation for the dependent variable. Denoting the basis functions as ξi and the weighting variable (also known as the degree of freedom) as Ui , the solution can be written for an element as  φ≃ Ui ξi (14.14) i

where the index i refers to the nodes in an element. The degrees of freedom are the desired solution variables in a finite element solution. In other words, one attempts to obtain the appropriate values of Ui that allow the constructed solution based on the summation of the basis functions ξi to match the actual solution of the governing partial differential equation. 14.2.3 Mesh Generation

The implementation of a finite element solution starts with the definition of a shape function and an element type. Typically, one discretizes a two dimensional

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NUMERICAL SIMULATION OF ELECTROKINETIC PHENOMENA

computational domain into triangular or quadrilateral elements. In the present formulation, triangular elements were employed with quadratic Lagrangian shape functions. The quadratic triangular elements typically have six nodes, three located at the vertices, and three at the mid-points of the triangle edges. The computational domain is discretized into non-uniform triangular elements, with smaller elements placed near the curved boundaries to ensure a more refined spatial discretization as shown in Figure 14.3. The mesh largely dictates the accuracy of the finite element solution. The mesh should adequately resolve the geometry, be sufficiently refined in regions where large gradients of the dependent variable exist, and should be of good “quality”. One needs to place a large number of elements in the regions where the boundaries are curved to capture the curvature adequately, such that geometrical errors arising from discretization is minimized. One also needs to have a refined mesh in regions where the dependent variable changes sharply (the spatial gradients of φ are large). Finally, the element quality is another important criterion in dictating the accuracy of the solution. The quality of an element, in simple terms, is a measure of how comparable the lengths of the three sides of a triangular element are; the similar the lengths are, the better the quality of the element. The degrees of freedom indicate the size of the matrix equation required in the solution for a given mesh. It depends on the number of elements, number of nodes in the element, number of governing equations, and the boundary conditions. The degrees of freedom dictate the size of the computational problem. The efficient solution of large matrix equations with minimal memory requirement constitutes the main algorithmic finesse of finite element solvers. For many problems, it is difficult to assess a priori which regions of the computational domain will have large spatial gradients, and hence, will require a refined mesh. Accordingly, it is common to employ mesh refinement strategies to converge to an accurate solution. Normally, most numerical calculations involve setting up an error criterion, and a tolerance level for that error. Then, the solution procedure is repeated with finer and finer meshes, until the tolerance is reached. Refinement of the mesh can

Figure 14.3. A typical initial finite element mesh consisting of quadratic triangular elements. The mesh has 10,299 elements and 21,416 degrees of freedom, with a minimum element quality of 0.53. Note that the elements are more refined near the curved surfaces of the particles and on the surface of the capillary wall. This is done to ensure that the curved surfaces and the regions where the potential decays more rapidly are adequately resolved.

14.2

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NUMERICAL SOLUTION OF THE POISSON–BOLTZMANN EQUATION

be performed by simply making every element smaller by a preset factor, a technique sometimes referred to as uniform refinement. However, this is computationally inefficient, and is not usually done in a commercial solver. These solvers frequently use adaptive refinement, where the mesh is refined locally in the regions where the spatial gradients are large. This type of adaptive mesh refinement is important in solution of Poisson–Boltzmann equation, since the spatial gradients of the electric potential are large in the vicinity of the charged surfaces, and refined meshes are often necessary in these regions. During refinement of mesh to obtain a more accurate solution, one does not have an absolute measure of the accuracy of the solution. The adaptation is usually done by comparing the results of two consecutive solutions of the same problem with two levels of mesh refinement. The difference between these two consecutive results then forms the basis for further mesh refinement. In the present problem, one of our goals is to obtain the net electric double layer force experienced by one of the particles in the capillary. Accordingly, we use this quantity as a metric for the mesh refinement. The calculation of the force will be discussed shortly. Table 14.1 depicts the manner in which the electric double layer force converges as the mesh is refined. The forces were obtained for a scaled separation κh = 0.4 and scaled particle size κa = 1.0, with a very large capillary radius κb ≃ 10. The results for both uniform and adaptive refinement are shown in Table 14.1. It is evident that the predicted force converges quite rapidly to a mesh independent value with adaptive mesh refinement as compared to uniform refinement, and requires considerably fewer number of elements in attaining the accuracy. This superiority of adaptive mesh refinement is ascribed to the ability of the technique to selectively place refined elements in the regions of the computational domain where more accuracy is required. 14.2.4 Solution Methodology Once the governing equations have been defined with appropriate boundary conditions, and the initial mesh has been generated, the weak form of the equations is written for each element. Incorporating information regarding how the various nodes

TABLE 14.1. Scaled Electric Double Layer Interaction Forces (see text for details) Obtained from Finite Element Calculations Employing Uniform and Adaptive Mesh Refinement. Uniform Refinement Step Number 1 2 3 4 5 6

Adaptive Refinement

No. of Elements

Force

No. of Elements

Force

212 848 3392 13568 54272 —

23.085 25.662 26.581 26.864 26.963 —

212 505 1153 2506 5496 11421

23.085 25.178 26.286 26.682 26.903 26.910

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NUMERICAL SIMULATION OF ELECTROKINETIC PHENOMENA

in adjacent elements are connected, a global set of algebraic equations is then developed. These coupled algebraic equations have the degrees of freedom (DOF), or the weighting parameters in Eq. (14.14), as unknowns. The degrees of freedom represent the number of coupled algebraic equations requiring simultaneous solution. Frequently this is a large number, and the resulting coefficient matrices are quite large. Furthermore, unlike finite difference techniques, these matrices may not be very ordered. In most finite difference formulations, the non-zero elements of the coefficient matrix are packed close to the diagonal elements. This is generally not the case for finite element problems. Finally, for non-linear problems, the matrix equations represent the linearized forms of the governing equation, and must be solved iteratively. Consequently, one has to solve large matrix equations employing specialized types of solvers during finite element analysis. This is one of the most memory intensive and computationally burdensome aspects of finite element calculations. This is also the reason why commercial solvers are so popular in finite element analysis. Most commercial finite element packages incorporate a robust matrix solver. These solvers can be of two types, namely, direct solvers and iterative solvers. Direct solvers are predominantly based on implementations of the Gaussian elimination procedure. Iterative solvers coupled with appropriate pre-conditioners are often less memory intensive than direct solvers. However, for highly non-linear problems, these techniques require a good initial guess for the solution. There are different algorithms for iterative solution, for instance, techniques based on Broyden search, generalized minimum residual (GMRES), and quasi-minimal residual (QMR). For the present problem, a direct solver was used to obtain the results. Starting from an initial mesh, the problem was solved incorporating adaptive mesh refinement. The tolerance in the global error was set to 10−6 . A maximum of 10 iterations in mesh adaptation was allowed. Finally, a limit of 30,000 for the maximum number of elements was imposed. Whenever one of the above criterion was met, the solution was terminated. The entire computation was performed in FEMLab. As mentioned earlier, the boundary conditions on segments AJ and HI (Figure 14.2) are artificial conditions, which may be valid only when these boundaries are located sufficiently far away from the particles. To ensure that the influence of these boundaries are negligible on the obtained solution, the simulations were performed by setting the length of AB and GH (Figure 14.2) to different values ranging from 1 to 10 times the particle radius. The resulting force calculations indicate that the artificial boundary conditions have negligible effect on the solution (computed force) when the lengths of the segments AB and GH are set to values ≥2κa. Accordingly, in all subsequent calculations, the boundaries AJ and HI were located at a distance 2.5 to 5 times the particle radius from the nearest particle surfaces. 14.2.5 Postprocessing: Calculation of the EDL Force The finite element solution provides the electric potential distribution over the computational domain. From this, the electric field components can be evaluated. One should note that to evaluate the EDL force on a particle, the components of the electric field (which constitute the Maxwell stress) must be obtained accurately from the solution

14.2

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NUMERICAL SOLUTION OF THE POISSON–BOLTZMANN EQUATION

of the Poisson–Boltzmann equation. Accordingly, the solution should provide highly accurate estimates of the electric field or the gradient of the potential. In the present case, a fairly accurate solution for the potential distribution could be obtained even using linear elements. However, since our focus was to obtain an accurate measure of the electric field, quadratic elements were employed in the solution. Taking this to a higher order accuracy, one could even have employed cubic elements, but higher order elements lead to more involved computation, and sometimes may even lead to an unstable solution. Decisions of this type are quite important in application of a numerical technique, and sound judgement in this regard is often acquired through experience. The force acting on a particle can be obtained by integrating the total stress tensor over the surface of the particle. Referring back to Chapter 5 (Section 5.4, see Example 5.8), the stress tensor at a given point on the particle surface is   = = = = 1 (14.15) T = −p I + Te = − p + ǫE · E I + ǫEE 2 =

=

Here pI is the hydrostatic (or osmotic) stress, Te is the Maxwell stress tensor, Eq. = (3.156), E (= −∇ψ) is the electric field, and I is a unit tensor. Integrating the stress tensor over the surface of a spherical particle yields the force       = = 1 (14.16) F = T · ndS = − p + ǫE · E I + ǫEE · ndS 2 S S Here, n is a unit outward normal to the particle surface. The integration should be performed over the particle surface S. It should be noted that the above expression for force is independent of the choice of the surface over which the integration is performed. In other words, the force can be computed by integrating the stress tensor over either particle surfaces BCD or EFG in Figure 14.2. Let us explicitly formulate the force for the present problem. From Eq. (5.105), it follows that for a symmetric (z : z) electrolyte, the osmotic (hydrostatic) stress contribution is given by p = 2n∞ kB T [cosh() − 1]

(14.17)

where n∞ is the bulk electrolyte concentration. Recalling from Chapter 5 that n∞ k B T ǫ = κ2 2



kB T ze

2

one can write the osmotic stress as 2

p = −κ ǫ



kB T ze

2

[1 − cosh()]

(14.18)

It should be noted that κ 2 ǫ(kB T /ze)2 has units of force per unit area (N/m2 ).

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NUMERICAL SIMULATION OF ELECTROKINETIC PHENOMENA

The Maxwell stress for the two dimensional system is given explicitly in a matrix form by   = = 1 ǫE · E I + ǫEE Te = − 2   1 0 1  2 2 = − ǫ (∂ψ/∂r) + (∂ψ/∂x) 0 1 2   (∂ψ/∂r)2 (∂ψ/∂r)(∂ψ/∂x) +ǫ (∂ψ/∂r)(∂ψ/∂x) (∂ψ/∂x)2 This expression can be written in terms of the non-dimensional potential ( = zeψ/kB T ) and the scaled coordinates (¯r = κr and x¯ = κx) as    

    = kB T 2 ∂ 2 ∂ 2 1 0 1 2 Te = κ ǫ + − 0 1 ze 2 ∂ r¯ ∂ x¯   (∂/∂ r¯ )2 (∂/∂ r¯ )(∂/∂ x) ¯ + (∂/∂ r¯ )(∂/∂ x) ¯ (∂/∂ x) ¯ 2 Combining the osmotic and Maxwell stresses, and simplifying the resulting expression, the total stress tensor, Eq. (14.15), can be written explicitly in a matrix form as      = r¯ x¯ kB T 2 (1 − cosh ) + 21 r¯2 − x2¯ 2 T=κ ǫ ze r¯ x¯ (1 − cosh ) + 21 (x2¯ − r¯2 ) (14.19) ¯ are shorthand notations for the where the terms r¯ (= ∂/∂ r¯ ) and x¯ (= ∂/∂ x) potential gradients (or electric field components) along r¯ and x, ¯ respectively. For the spherical particle BCD in Figure 14.2, which is centered at the origin of the coordinate system, the unit outward surface normal can be written as n = nr¯ ir¯ + nx¯ ix¯ r¯ x¯ =√ ir¯ + √ ix¯ 2 2 2 r¯ + x¯ r¯ + x¯ 2

(14.20)

Using the expressions of the total stress tensor, Eq. (14.19) and the unit surface normal, Eq. (14.20) in Eq. (14.16), one obtains       1 − cosh  + 21 r¯2 − x2¯ nr¯ + r¯ x¯ nx¯ kB T 2 2  F=κ ǫ dS (14.21) 1 2 2 ze S r¯ x¯ nr¯ + 1 − cosh  + 2 (x¯ − r¯ ) nx¯

Note that the term within the integral represents a vector quantity, with the upper line indicating the force component acting along the r¯ coordinate, and the lower line representing the force component along x. ¯

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553

Since we are interested in the component of the force acting along the axis of the cylinder (x), ¯ we obtain this force as 2  

 1 2 2 r¯ x¯ nr¯ + 1 − cosh  + (x¯ − r¯ ) nx¯ dS Fx¯ = F · ix¯ = κ ǫ 2 S (14.22) Now, for the spherical particle denoted by the arc BCD, the area elements can be defined in the present coordinate system as dS = 2π r¯ d x/κ ¯ 2 . Substituting this in Eq. (14.22), one obtains, 2



kB T ze

2 



 1 2 2 Fx¯ = 2π ǫ r¯ x¯ nr¯ + 1 − cosh  + (x¯ − r¯ ) nx¯ r¯ d x¯ 2 BCD (14.23) Equation (14.23) represents the net interaction force acting along the axial coordinate (x) ¯ on the particle BCD in Figure 14.2. Referring to Figure 14.2, the force will be acting toward the left (along the negative x direction) if it is repulsive, while it will be acting toward the right (along positive x direction) if there is attraction between the two spherical particles. We note that representing the force in the form of Eq. (14.23) allows the use of the scaled potentials and their spatial gradients in the same form as employed during the finite element solution. Finally, the force can be expressed in a non-dimensional form as   Fx¯ ze 2 fx¯ = (14.24) ǫ kB T 

kB T ze





In the remainder of the discussion in this section, we will use this non-dimensionalized form of the force. It should be noted that one can obtain an analogous expression for the force on the particle EFG. While the expression for the force will be somewhat different from Eq. (14.23), the magnitude of the force calculated on either BCD or EFG will be same. This is a very pertinent self-consistency check of the computation, and should be carried out to ensure the accuracy of the numerical program. The integration of the stress tensor according to Eq. (14.23) can be conducted employing the built-in postprocessing unit of the finite element software. The boundary integration procedure adopted for this purpose employs Gaussian quadrature, which provides a highly accurate result with few integration points. 14.2.6 Validation of Numerical Results A validation of the finite element solution against existing results can be obtained for the case of two spherical particles interacting in a large electrolyte reservoir. The interaction between two constant potential or constant charge spheres is a well-studied problem, and numerous analytical and numerical solutions exist for this problem. In the present formulation, the problem of two particles interacting inside a cylindrical capillary can be modified to obtain a solution for the two interacting spheres. This can be performed by making the capillary radius b considerably larger than the particle

554

NUMERICAL SIMULATION OF ELECTROKINETIC PHENOMENA

radii, a, and making the capillary surface neutral. The capillary surface can be made neutral by simply assigning a boundary condition of the form ¯ =0 n · ∇

on IJ

(14.25)

One should, however, note that this artificial condition can only be applied when b ≫ a. Employing these conditions, one can obtain the interaction force between two spheres separated by a distance h without having to re-formulate the problem. The electrostatic force on a particle determined from the finite element analysis for this limiting case can be compared with independent estimates of the force available in literature. A numerical solution procedure for the Poisson–Boltzmann equation was developed by Carnie et al. (1994), which utilized a bispherical coordinate system. They employed a cubic Hermite collocation based technique for discretization of the governing equation, followed by an iterative solution of the resulting non-linear system of equations employing a Newton–Raphson technique to obtain the interaction force. This numerical solution provides very accurate estimates of the electrostatic interaction force between two spherical particles. The solution is applicable for any surface potential on the particles, and for a wide range of separation distances and the parameter κa. One should recall from our earlier discussions that the numerical solutions tend to falter at large values of the parameter κa. However, these solutions are accurate when κa ≤ 10. Exact analytical expressions for the sphere-sphere interaction force based on a solution of the non-linear Poisson–Boltzmann equation are not available. There are several results based on the solution of the linearized Poisson–Boltzmann equation. These range from the simple result of Hogg et al. (1966), which is based on Derjaguin’s approximation, to some complex series solutions based on a multipole expansion technique (Glendinning and Russel, 1983). Here, we will compare the numerical results obtained from the finite element calculations with the results of Carnie et al. (1994), as well as, with the simple analytic result of Hogg et al. (1966). Figure 14.4 shows a comparison between the scaled interaction force, fz , obtained from the finite element calculations, and those obtained by applying the Hermite collocation solution of Carnie et al. (1994). The agreement between the two solutions is excellent, given that they were obtained using completely different coordinate systems, and two different numerical techniques. However, the good agreement is expected, since we are essentially solving the same physical problem. What makes the agreement remarkable is that in the finite element calculations, the cylinder wall radius was chosen to be only five times the particle radius, whereas the solution in the bispherical coordinate system employs the true boundary condition at infinity. There is another difference between the two calculations. The interaction force on a sphere was computed in the finite element calculations using an integration of the stress tensor over the particle surface. In the Hermite collocation solution, the interaction force was computed by integrating the stress tensor at the mid-plane between the two particles. Integration of the stress tensor over the mid-plane makes considerably more sense when one employs the bispherical coordinate system. However, this is inadequate for the geometry employed in the finite element calculations.

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Figure 14.4. Comparison of the electric double layer force between two spherical particles obtained using the finite element analysis (symbols) with the calculations of Carnie et al. (1994) (lines) for b ≫ a and a neutral capillary surface. (a) Results for constant surface potential (CP) on the particles. (b) Results for constant surface charge density (CC) on the particles. In case (b) the surface charge density on an isolated particle was computed from the given value of p,∞ . Details of this comparison is available in Das et al. (2003).

The agreement between the forces calculated on two surfaces clearly indicates that integration of the stress tensor over any closed surface enclosing a sphere will provide the total interaction force. In case of the bispherical coordinate system, the integration over the mid-plane actually represents an infinite surface enclosing one of the spheres. The comparison of the forces obtained from the two procedures indicates the ability of finite element approximation to provide a reasonable estimate of the EDL interaction forces. This, however, comes at the expense of fairly involved computation. It should be noted that the good agreement between the two force estimates could not be attained without using adaptive mesh refinement, employing a large number of elements (generally 20,000 to 30,000), and carefully choosing the convergence criterion. Furthermore, one should note that all the boundary conditions employed in the finite element model for the sphere-sphere interaction, except for those on the particle surfaces, are artificial. If one plans to solve the Poisson–Boltzmann equation to obtain the interaction force between two spheres in an infinitely large domain, use of bispherical coordinates is definitely the most appropriate route. Nevertheless, the accuracy of the finite element calculations for this limiting case does validate the correctness of the numerical solution procedure, and provides confidence regarding its applicability in cases when the effect of the charged capillary wall on the interaction force is calculated.

556

NUMERICAL SIMULATION OF ELECTROKINETIC PHENOMENA

In Figure 14.5, the scaled interaction force predicted by the finite element calculations are compared with the scaled EDL force computed by the expression given by Hogg et al. (1966), often referred to as the Hogg–Healy–Fuerstenau (HHF) expression. The HHF expression for the interaction force between two identical spheres is given by Eq. (13.70) for constant surface potential on the spheres. Note that for identical surface potentials, the double layer asymmetry parameter, Da, (cf., Eq. 13.73) will be zero, resulting in a somewhat simplified form of the equation. One should note that the expression of Hogg et al. (1966) is applicable only when the surface potentials on the particles are small (strictly speaking, zeψp /kB T ≪ 1), and for large particles (κa > 10). The above limitations of the HHF result stems from the use of linearized Poisson–Boltzmann equation and Derjaguin’s approximation. Nevertheless, this expression has been used to obtain the interaction forces between particles that have much larger surface potentials, and sometimes even for κa ≃ 2). The comparison of the two calculations in Figure 14.5 (left) indeed shows that at low surface potentials, p = −1, the analytical HHF result provides fairly accurate estimates of the interaction force for κa ≃ 5. The deviation of the HHF result from the finite element predictions increases at lower values of κa. In these cases, the HHF expression underpredicts the interaction force at small separations, but overpredicts the interaction force at large separations. Figure 14.5 (right) shows the comparison between the analytic and numerical results for κa = 5 corresponding to different values of surface potentials. At higher surface potentials, the analytical result deviates from the numerical predictions of the interaction force considerably. The deviations indicate the extent of error introduced in the force estimates due to linearization of the

Figure 14.5. Comparison of the electric double layer force between two spherical particles obtained using the finite element analysis (Symbols) with the analytic solution of Hogg et al. (1966) (lines). Left: Comparison for different κa for low surface potential, p = −1. Right: Comparison for κa = 5 and different values of surface potential.

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557

Poisson–Boltzmann equation, and the use of Derjaguin’s approximation embedded in the HHF result. The above comparisons of the finite element estimates of the interaction force serve as validation of the numerical code against known solutions of the Poisson–Boltzmann equation. This validation procedure is essential in any numerical calculation. Once the numerical methodology is validated, it can be employed to explore the dependence of the interactions on different parameters. 14.2.7 EDL Interaction Force on Particles in a Charged Capillary The interaction force between two spherical particles can be considerably modified when the particles are trapped in a capillary of comparable radius. Detailed analysis of the interaction forces experienced by the spherical particles in a capillary with different surface potentials has been recently reported (Das et al., 2003). Figure 14.6 depicts the variation of the interaction force with separation distance between the particles for different scaled surface potentials, c , on the capillary wall. The scaled particle surface potential is fixed at p = −3, and the aspect ratio, λ = b/a = 1.2 for all the calculations. In this figure, the symbols represent the scaled interaction force between the spheres in an unbounded electrolyte solution. When the capillary radius is comparable to the particle radius and the capillary surface potential has the same sign and magnitude as the particle surface potentials, there is a significant reduction in the electric double layer repulsion between the particles. In no case, however, an

Figure 14.6. Scaled electric double layer interaction force, fx¯ , between two spherical particles inside a cylindrical charged capillary (lines). In all cases, the particles have a constant scaled surface potential, p = −3. The capillary surface potential, c , is varied to study its influence on the particle-particle interaction force. The ratio of the capillary radius to the particle radius, λ = b/a, is fixed at 1.2. The symbols represent the interaction force between the spheres in an infinite reservoir in absence of the charged capillary wall (Das et al., 2003).

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attractive EDL force is observed for such a geometry, as long as the capillary wall remains smooth. The influence of the capillary wall on the interaction force is most pronounced for small values of the parameter κa ≃ 1 [Figure 14.6(a)]. For larger values of this parameter (κa = 5), there is virtually no influence of the capillary wall on the interaction force between the particles [Figure 14.6(b)]. Das et al. (2003) also observed that the EDL repulsion between the particles is enhanced when the capillary wall surface potential has an opposite sign to the particle surface potentials. Similar calculations can also be performed by employing constant charge boundary conditions on the surface of the particles and the capillary wall. Although presence of the capillary wall has some influence on the interaction force between constant charge particles, the modification of the force is not as significant as in the case of constant potential particles. Simulations for the interaction between particles and capillary walls having mixed boundary conditions (constant charge on one particle while constant potential on the other) can also be performed. The presence of roughness on the capillary wall considerably influences the interaction force between the particles. Simulations of these different scenarios have also been reported (Das and Bhattacharjee, 2004, 2005). The presence of roughness on the capillary wall modifies the electric field distribution at different locations of the capillary wall surface. Such local modifications can give rise to different forces on each particle. In such situations, one may even observe an attractive force between two similarly charged particles. An interesting observation regarding the interaction force components, namely the osmotic stress and the electrical (Maxwell stress) is that for a constant potential particle, the osmotic stress is equal at all locations on the surface (cf., Eq. 14.17). Consequently, integration of this stress component over the closed surface of the particle makes the osmotic pressure contribution to the total interaction force zero. Therefore, the entire force modification observed for constant potential particles is due to modifications in the Maxwell stresses. For constant charge particles, however, the surface potential on the particle will be different at different locations. Consequently, the integration of the osmotic stresses over the particle surface will result in a finite osmotic pressure force. The analysis of the Poisson–Boltzmann equation described in this section should not be considered a more accurate representation of the actual physics of electrostatic interactions. All the approximations inherent in the development of the Poisson– Boltzmann equation are still present in the numerical results, and hence, the influence of the various assumptions, for instance, the point charge ions, continuum solvent, etc., on the result need to be assessed. Noting that the solution was obtained for small values of the parameter κa ≃ 1, this would imply that for aqueous systems, the capillary radius is of the order of ten nanometers for salt concentrations of about 10−4 M. This may seem to be too small a pore radius to allow a continuum analysis. However, small κa values can also be achieved with fairly large particle radii in solvents with lower dielectric constants than water. The developed numerical solution is therefore also of interest when dealing with non-aqueous, low dielectric constant solvents. For such systems, it is evident from the above results that the electric double layer force between two spherical colloidal particles can be considerably modified due to the presence of the capillary wall.

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In this section, the numerical procedure for solving the Poisson–Boltzmann equation was presented in considerable detail. In particular, the rendition of the physical problem to a finite element formulation was elaborated in some detail. In summary, the Poisson–Boltzmann equation represents a single stationary (time independent) partial differential equation. From a numerical standpoint, solution of such an equation should be fairly tractable. However, the correct accounting of the physics in the mathematical formulation required considerable attention to detail, particularly concerning the placement of boundary conditions. In the following sections, we will address two problems that are more complicated from a numerical standpoint.

14.3 FLOW OF ELECTROLYTE IN A CHARGED CYLINDRICAL CAPILLARY In this section, we will discuss numerical simulations of the combined pressure and electric potential gradient driven electrokinetic flow of an electrolyte solution through a cylindrical capillary microchannel having a charged wall. This problem was discussed in Chapter 8 as electrokinetic flow in capillary microchannels. Electrokinetic flow is of significant interest in microfluidic devices (Li, 2004). During pressure driven flows through microchannels, a streaming potential is developed across the capillary due to the flow of the ions relative to the stationary charged walls of the channel. The transport of an electrolyte solution through a capillary microchannel can be treated as either a steady-state or a transient problem. The model requires a coupled solution of three sets of governing equations representing the coupled effects of three physical processes. First, the distribution of the electric potential in the system is governed by the Poisson equation. Secondly, the flow of the electrolyte solution is dictated by the fluid continuity and Navier–Stokes equations. Finally, the transport and distribution of the ions are governed by the Nernst–Planck equations. These three types of governing equations must be solved simultaneously to obtain the electric potential distributions, the flow velocities and fluxes, and the ion concentration distributions in the system. The theory of electrokinetic transport, particularly analyses pertaining to the calculation of streaming potential, has been thoroughly developed over the past century, with numerous studies exploring different aspects of the governing transport phenomena (Helmholtz, 1879; Smoluchowski, 1903; Lewis, 1960; Osterle, 1964; Burgreen and Nakache, 1964; Rice and Whitehead, 1965; Morrison, 1969; Levine et al., 1975; Yang et al., 2001; Keh and Tseng, 2001; Daiguji et al., 2004). Most of these studies are based on a steady-state analysis of the governing electrochemical transport equations applied to microchannels of infinite length. These studies, apart from those of Morrison (1969) and Keh and Tseng (2001), do not address issues associated with the transient development of the electric potential across the microchannel. Furthermore, barring a handful of studies (Yang et al., 2001; Daiguji et al., 2004) the entry effects, or the effects of charged walls of the reservoirs connecting the capillary are rarely considered.

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Electrokinetic flow in narrow capillary channels has also received considerable attention in the context of transport of electrolytes through reverse osmosis and ultrafiltration membranes (Gross and Osterle, 1968; Jacazio et al., 1972; Anderson and Malone, 1974; Sasidhar and Ruckenstein, 1981, 1982; Deen, 1987; Cwirko and Carbonell, 1989). It is well known that transport of an electrolyte through membrane pores results in a rejection of the ions, yielding a lower electrolyte concentration in the solution emerging from the pore. The space charge model was an outcome of the initial attempts at analyzing the transport of ions through narrow porous media. More recently, a considerable body of studies employing the so-called one-dimensional extended Nernst–Planck equations (Smit, 1989; Basu and Sharma, 1997; Hall et al., 1997; Bhattacharjee et al., 2001) have been used to study ion transport and rejection by membrane pores. Once again, most of these modeling approaches are based on steady-state analysis. Electrokinetic flow in microchannels and transport through membrane pores are governed by the same fundamental equations, namely, the Navier–Stokes equations for the fluid flow, the Nernst–Planck equations for the ion transport, and the Poisson equation for the electric potential and field distributions. Both types of transport problems essentially attempt to emulate a physical picture shown in Figure 14.7, where a capillary connects two reservoirs containing bulk electrolytes. Note that in Figure 14.7, the outer boundaries of the reservoirs (shown by dashed lines) simply represent locations in the reservoir that are sufficiently far away from the capillary to allow the assumption of bulk conditions like electroneutrality to hold. Depending on the

Figure 14.7. Schematic diagram of a capillary microchannel connecting two reservoirs. When a pressure gradient is set up between the two reservoirs, a potential difference can be recorded between the electrodes placed in the reservoirs. The potential difference under steady-state flow conditions is referred to as streaming potential. Alternately, setting up a potential difference between the electrodes will induce an electroosmotic flow through the capillary in absence of any applied pressure gradient.

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nature of the problem, for instance, whether one intends to determine the streaming potential during electrokinetic flow, or assess salt rejection during membrane transport, the conditions in the outlet reservoir are stated differently. In streaming potential analysis, the electrolyte concentrations in the inlet and outlet reservoirs are held fixed at same values. In problems dealing with prediction of salt rejection by capillary microchannels, the electrolyte concentration in the outlet reservoir is not known a priori. Consequently, one needs to specify an alternate set of conditions for this reservoir. The analytical models for electroosmotic flow in Chapter 8 were developed without explicitly considering the two reservoirs at the two ends of the capillary microchannel. The analytical model simply applied to a section of an infinitely long capillary, and neglected any effects of the entry and exit flows at the capillary ends. The fluid flow inside the capillary was assumed to be strictly one dimensional. Adopting a suitable numerical procedure to solve the governing equations for the geometry of Figure 14.7 can lead to an assessment of the deviations in streaming potential engendered by the approximations inherent in the analytical models. In particular, a numerical procedure for solving the governing equations can provide detailed information regarding the ion concentration distributions in the channel and the reservoirs, the velocity distribution, and the electric potential distribution. Let us first summarize the implications of the different approximations inherent in the analytical models for electrokinetic flow. Typically, these models assume an infinitely long capillary, and ignore any axial (along the length of the capillary) variation of ion concentrations (Rice and Whitehead, 1965). Such assumptions allow the use of an equilibrium radial ion concentration distribution in the capillary, which is obtained from the solution of the one-dimensional Poisson–Boltzmann equation. Most analytical solutions, and sometimes numerical simulations, of electrokinetic transport processes are obtained employing the undisturbed ion concentration profiles based on the Poisson–Boltzmann equation (see, for instance, the theoretical development in Chapter 8 for electroosmotic flow in capillary microchannels). The obvious limitation of such an approach is its inability to predict any axial variation of ion concentrations, and hence, any rejection of ions by the capillary. In contrast, the extended Nernst–Planck models applied to study ion transport through membrane pores predict a substantial axial variation of the ion concentrations in the capillary, and hence, salt rejection. The steady state analytical models also fail to provide a clear assessment of the time scales required for development of the steady-state streaming potential. According to some electrokinetic theories (Morrison, 1969; Keh and Tseng, 2001), the steady state streaming potential is attained within the time scale of the hydrodynamic relaxation. In contrast, the extended Nernst–Planck approach applied to pore transport problems predicts establishment of the steady state over the time scale of diffusion of the ions. The diffusion time scale is several orders of magnitude slower than the hydrodynamic time scale. This leads to the question as to what is the true time scale for attainment of steady state during electrokinetic flow through narrow capillaries? The above mentioned apparent discrepancies between the electrokinetic flow and membrane transport models can be systematically resolved through a coupled solution

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of the transient electrochemical transport equations for the geometry depicted in Figure 14.7. More specifically, the geometry obviates the requirement of specifying the boundary conditions directly at the capillary entrance and exit. Furthermore, a transient solution of the governing equations can provide considerable insight regarding the time scales of the development of the steady state flow characteristics. Using such a geometry, Mansouri et al. (2005) studied the ion transport behavior in a finite length capillary, with different types of boundary conditions imposed at the inlet and outlet reservoirs. Their analysis was primarily restricted to the flows engendered by an applied pressure gradient along the capillary. Here we generalize the model to accommodate simultaneous application of the pressure and electric potential gradients along the capillary.Accordingly, solutions for the general electroosmotic flow problem occurring under the combined influence of pressure and electric potential gradients across the capillary channel will also be discussed in this chapter. Thus, within the scope of the same geometrical description and the same governing equations, we address three separate case studies pertaining to electrokinetic flow. These are: Case 1: Prediction of streaming potential developed at steady-state during pressure driven flow through a charged capillary microchannel connecting two reservoirs. In this case, the two reservoirs contain electroneutral electrolyte solutions of identical concentrations. Case 2: Prediction of the transient evolution of steady-state electric potential and ion concentration distributions in a capillary microchannel during pressure driven flow. In this case, the electrolyte concentration in the outlet reservoir is unknown, thus emulating the problem of ion rejection through a membrane pore. Case 3: Prediction of the steady-state fluid axial velocity in a capillary microchannel under the coupled influence of pressure and electric potential gradient. 14.3.1 Problem Formulation The computational geometry and the governing equations common to all three case studies mentioned above are presented here. The numerical solution procedure for the three case studies primarily differ in the manner in which the boundary conditions are specified. Accordingly, we will first present all aspects of the numerical solution procedure that are common to all three cases, following which, we will separately discuss the boundary conditions of the three case studies. The model geometry represents a cylindrical capillary connected to two reservoirs, as shown in Figure 14.8. The entrance and exit reservoirs are modeled as cylindrical manifolds of radius b, which is much larger than the capillary radius, a. Utilizing the axial symmetry of the geometry, all the governing equations will be written in axisymmetric cylindrical coordinates (r, x) to simplify the model to two dimensions. The axisymmetric model geometry is represented by the region bounded by the outer boundary ABCDEF, and the line of axial symmetry, PQ. The dashed boundaries AP, AB, EF, and FQ represent the regions in the reservoirs that are unaffected by the capillary. These boundaries are either represented through bulk conditions or appropriate symmetry conditions. The distances of these boundaries from the entrance and exit

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Figure 14.8. Computational geometry for modeling electrokinetic flow of an electrolyte in a cylindrical capillary.

regions of the capillary should be sufficiently large to ensure that the electrochemical properties at these locations are not influenced by the capillary. The model geometry also incorporates the walls of the inlet and outlet reservoirs (BC and DE). One can assign appropriate electric potentials or charge densities on these walls or simply render them as electrically neutral surfaces. The wall of the capillary (CD) can bear a constant electric potential or a constant charge density. All the solid surfaces are assigned no slip boundary condition, and are assumed to be impermeable to the ions. In all the three case studies, we consider the transport of a symmetric (z : z) binary electrolyte solution of a specified bulk molar concentration, c∞ . In Cases 1 and 2, the flow is driven solely by an externally applied pressure gradient. In Case 3, an external electric field is also applied in the axial direction, causing a flow influenced by the combined effects of pressure and electric potential gradients. Various dimensions and parameter values pertaining to the model are shown in Table 14.2. The ionic diffusivity is assumed to be equal for all ions. The ratio of the radii b to a is referred to as the capillary expansion factor, and is assigned a value of 5 in all subsequent discussion. The capillary length is chosen to be larger than the capillary radius by a factor of 10, so that the entrance and exit effects on the overall transport phenomenon is small. The electric potential distribution in the electrolyte solution is related to the volumetric free charge density by Poisson’s equation: ∇ 2ψ = −

ρf ǫ

(14.26)

Here ǫ is the permittivity of the fluid, ψ is the prevailing electric potential, and ρf is the free charge density, given by the equation  ρf = zi eni (14.27) i

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TABLE 14.2. Parameter Values Employed in the Model for Flow of an Electrolyte in a Charged Capillary. Parameter

Value/Range

Solvent permittivity, ǫ Capillary wall potential, ψc Bulk electrolyte concentration, c∞ Inverse Debye length, κ Ion valence, zi Ion diffusivity, D Temperature Fluid density, ρ Fluid viscosity, µ Axial pressure gradient, px = −∂p/∂x Capillary radius, a Capillary expansion factor, b/a Capillary length

78.54 × 8.854 × 10−12 C2 /Nm2 0 to −75 mV 10−5 M 1.038 × 107 m−1 1 1 × 10−9 m2 /s 298 K 1000 kg/m3 0.001 N.s/m2 107 Pa.m−1 κ −1 –10κ −1 m 5 10κ −1 m

where zi and ni are the valence and number concentration of the ionic species, respectively. The movement of the ions in the system is governed by the Nernst–Planck equations. For each ionic species, the Nernst–Planck equation can be written as   ∂ni zi eni D ∗∗ = −∇ · ji = −∇ · ni u − D∇ni − ∇ψ ∂t kB T

(14.28)

−2 −1 where j∗∗ i is the ionic number flux (m s ), u is the fluid velocity vector, and D is the diffusion coefficient. If we consider only the steady-state solution, Eq. (14.28) becomes   zi eni D ∇ψ = 0 (14.29) ∇ · ni u − D∇ni − kB T

To determine the velocity field, the Navier–Stokes equation is used. The Navier– Stokes equation, including a volume force due to an electric field, is given by ρ

∂u + ρu · ∇u = −∇p + µ∇ 2 u + ρg − ρf ∇ψ ∂t

(14.30)

Here ρ and µ are the fluid density and viscosity, respectively, and p is the pressure. Gravitational force can be neglected due to the small scale of the problem. The electrical body force per unit volume is given by the product of the charge density ρf and the local electric field E = −∇ψ. When the steady-state solution of the problem is considered, removing the time dependent term from Eq. (14.30) yields, ρu · ∇u = −∇p + µ∇ 2 u + ρg − ρf ∇ψ

(14.31)

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TABLE 14.3. Dimensionless Variables Employed to Scale the Governing Equations. Dimensionless Variable

Expression

Radial coordinate, r¯

κr

Axial coordinate, x¯

κx

Time, τ

κ 2 Dt

Velocity, U

u/κD

Ion concentration, Ni

Surface charge density, σc

ni /n∞ zeψ kB T ze E kB T κ ze qc ǫkB T κ

Fluid density, ρ¯

z2 e 2 D 2 ρ ǫkB2 T 2

Viscosity, µ¯

z2 e 2 D µ ǫkB2 T 2

Pressure, P

z2 e 2 p ǫkB2 T 2 κ 2

Electric potential,  Electric field, E¯

For an incompressible fluid, the Navier–Stokes equation is solved along with the continuity equation given by ∇ ·u =0

(14.32)

In order to obtain the governing equations in dimensionless form, we will consider the case of a symmetric electrolyte, and scale all dimensions with respect to the inverse Debye length κ=



2n∞ z2 e2 ǫkB T

1/2

(14.33)

where n∞ is the bulk electrolyte concentration. By introducing the dimensionless variables given in Table 14.3, the governing equations can now be stated in their scaled forms. The Poisson equation in the non-dimensional form is given by   Np − Nn 2 ¯ ∇ =− 2

(14.34)

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where Np and Nn are the dimensionless concentrations of the positively and negatively charged species, respectively. Here, the scaled gradient operator ∇¯ is defined as ∇¯ = κ −1 ∇ =



∂ ∂ ir¯ + ix¯ ∂ r¯ ∂ x¯



(14.35)

where ir¯ and ix¯ are the unit vectors along the scaled radial and axial coordinates, respectively. Scaling the Nernst–Planck equation gives the following equations for the transport of positive and negative ions, respectively, ∂Np ¯ p − Np ∇] ¯ = −∇¯ · [Np U − ∇N ∂τ ∂Nn ¯ n + Nn ∇] ¯ = −∇¯ · [Nn U − ∇N ∂τ

(14.36-a) (14.36-b)

Finally, the Navier–Stokes in absence of gravity and continuity equations are also scaled to obtain   N p − Nn ¯ ∂U 2 ¯ ¯ ¯ + ρU ¯ · ∇U = −∇P + µ¯ ∇ U − ∇ (14.37) ρ¯ ∂τ 2 and ∇¯ · U = 0

(14.38)

Note that in the scaled form of the Navier–Stokes equation, the gravitational body force term was dropped. Equations (14.34) to (14.38) represent the dimensionless forms of the governing equations for the general electrokinetic flow problem pertaining to all three case studies that will be conducted in this section. These equations were written retaining the time dependent terms. Dropping the time derivative terms from the Nernst–Planck and Navier–Stokes equations will provide the corresponding steady-state transport model. Although these equations seem to be quite general, they are based on several assumptions, for instance, symmetric electrolyte (zp = −zn = z), same diffusivity for all the ionic species, fixed solution density, and constant viscosity. Furthermore, the diffusivities of the ions will generally be much smaller in narrow capillaries than the corresponding values of the parameter in bulk solutions. Most of these assumptions can be generalized. However, to render the numerical model identical to the analytical problems described in Chapter 8, these assumptions were retained in the governing equations. It should be noted that all the linear dimensions in the governing equations were scaled with respect to the screening length (Debye length) of the electric double layer. This was done deliberately to explore the electroosmotic phenomena in capillaries that are comparable in dimensions to the screening length of the electric double layer. As mentioned earlier in this chapter, most analytical theories of electroosmotic flow were developed for cases when the channel dimensions are significantly larger than the Debye length. Typically, for aqueous systems containing a monovalent salt

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at a reasonable concentration flowing through a large radius capillary, the parameter κa, where a is the channel radius, is often >100. Attempting a numerical solution for such large values of the parameter κa should not, in principle, provide any further insight compared to analytic solutions, such as the Helmholtz–Smoluchowski result. In this context, application of numerical techniques can be justified when addressing electroosmotic flows in narrow capillaries, where the analytic expressions derived for the limiting cases of large κa ≫ 1 might provide inaccurate results. 14.3.2 Mesh Generation and Numerical Solution The mesh generation and the numerical solution procedure are similar for all the case studies. Hence, we present these components of the numerical modeling procedure prior to the individual description of these case studies. The computational domain bounded by ABCDEFQP in Figure 14.8 was discretized into quadratic triangular elements. For the present problem, there are three governing models dictated by three sets of equations. A complete adaptive mesh refinement scheme that allows minimization of numerical errors for all three sets of governing equations is quite difficult and memory intensive. Consequently, we solve the problem with a predefined mesh containing at least 10,000 elements. A typical mesh is depicted in Figure 14.9. The mesh consists of about 25,000 elements, with smaller elements placed along the capillary and reservoir walls, as well as the line of axial symmetry. Since adaptive mesh refinement could not be used in the computations, the problem was solved with meshes having different numbers of elements, and the solutions obtained for different mesh sizes were compared to ensure that mesh independent results were obtained. Most of the simulation results became mesh independent when the number of elements was about 15,000.

Figure 14.9. Finite element mesh used to solve the electrokinetic flow problem. A close-up view of the mesh is shown over the boxed region of the computational domain.

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The finite element software FEMLab provides a simple and intuitive interface for modeling the above multiphysics problem. The software provides individual modules for solving the Poisson, Navier–Stokes, and Nernst–Planck Equations. A separate problem module is set up for each governing physics. One selects the governing equation to be solved for the geometry, defines the parameters of these equations, along with the appropriate initial and boundary conditions, and ensures that the coupled variables are denoted similarly in different equation modules. For instance, one should use the same parameter name, , for the scaled electric potential in the Poisson, Nernst–Planck, and the Navier–Stokes modules. Similarly, the concentration of the positive and negative ions are denoted by Np and Nn , respectively, in all the three equation modules. Availability of these predefined modules renders the setting up of the multiphysics problem quite facile. Note that in this case, we are not explicitly required to recast every governing equation to the general PDE form, as was done in Section 14.2. Instead, the selected module for the governing equation in FEMLab reads in the parameters for the actual equation, and performs the conversion of the governing equation to the required general or weak forms internally. Most of the numerical methodology validation, and subsequent calculations were done using a value of κa = 5, which means that the capillary radius is five times the Debye length. The radius of the inlet and outlet manifolds, b, was five times the capillary radius. The scaled length of the capillary (CD) was 50, giving a length to diameter ratio of 5. Finally, the scaled length of the inlet and outlet reservoirs (AB and EF) was 25, yielding a total length of the capillary and reservoirs (PQ) of 100. The program was set up to solve either the steady-state or the transient equations. The transient governing equations were solved from an initially no flow condition until the steady-state flow conditions were attained after imposition of an axial pressure and/or an applied axial electric potential gradient. The numerical scheme for the solution of the transient equations for simulating the flow under an applied pressure gradient is depicted in Figure 14.10. After defining the problem parameters, the geometry, and creation of the mesh, we first solve the Nernst–Planck and Poisson equations in absence of any fluid flow. This yields the Poisson–Boltzmann stationary ion and electric potential distribution over the computational domain. Once this initial distribution is calculated, a pressure gradient is employed along the capillary, and the transient governing equations are solved iteratively starting from the initial quiescent solution. Using the Poisson–Boltzmann ion and concentration distributions, the free charge density and electric field terms appearing in the body force of the Navier– Stokes equation are first determined. The velocity field is then obtained by solving the Navier–Stokes equation. The velocity field is substituted in the Nernst–Planck equations to obtain an updated estimate of the ion concentration distributions in presence of convection. Substituting these ion concentration distributions in the Poisson equation then provides the electric potential distribution at the next time step. The updated electric potential and charge density distributions are then substituted back in the Navier–Stokes equations to provide the new velocity field. This iterative solution procedure is continued until the solution at two consecutive time steps converges to within a preset tolerance. The solution procedure was implemented on the software FEMLab employing its in-built solver manager. The solver manager runs the solution

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Figure 14.10. Flowchart depicting the finite element solution methodology for the streaming potential development problem. The region demarcated within the dashed rectangle represents the iterative solution procedure for the transient problem.

procedure using a script which automates the sequential solution of the three sets of governing equations. The solution procedure depicted in Figure 14.10 addresses the development of the flow under the sole influence of an applied pressure gradient (which is pertinent to the first two case studies). To solve the complete electroosmotic flow problem in presence

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of both an applied pressure gradient and an external electric field (Case 3), one only needs to modify the electrical boundary condition (for the Poisson equation) at the flow outlet, where an external electric field E0 , is imposed as a boundary condition. Even though the governing equations, the computational domain and mesh, as well as the general solution methodology remain unchanged in all subsequent analysis, application of different boundary conditions alter the overall problem formulation to reflect the three case studies we intend to conduct. We will discuss these three cases separately in the following. 14.3.3 Case 1: Streaming Potential Across a Capillary Microchannel In this case study, we simulate the steady-state streaming potential developed between two reservoirs connected by a narrow capillary microchannel. This case was treated analytically in Chapter 8. Here we describe how one can perform the numerical simulations to obtain results that emulate those predicted by the analytical approach. 14.3.3.1 Boundary Conditions To define the boundary conditions for the steady-state transport problem, we focus on the computational domain in the two dimensional cylindrical coordinate system (r, x) as shown in Figure 14.11. Several of the boundary conditions remain unaltered in all three case studies. For instance, PQ is the line of axial symmetry (axis of the capillary microchannel). Accordingly, all the governing equations will have the axial symmetry condition on the segment PQ for all three case studies. The boundary conditions on the segments AB and EF will also be similar in all three problems. We will only change the conditions at the charged walls (BC, CD, and DE) as well as on the line segments AP and FQ to differentiate between the problems we are solving. The appropriate boundary conditions applicable for each governing equation that are pertinent to the streaming potential analysis are described in Table 14.4. As stated earlier, axial symmetry exists for all the governing equations on segment PQ. The boundary conditions on the segments AB and EF are defined assuming these surfaces to be in the bulk electrolyte reservoirs. Accordingly, for the Poisson equation, we will

Figure 14.11. Schematic representation of the boundary conditions for the Navier–Stokes, Nernst–Planck, and Poisson equations used in the simulation of streaming potential (Case 1). On the symmetry planes AB and EF, the ion concentration gradients and potential gradients normal to the planes are assumed to be zero.

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TABLE 14.4. Boundary Conditions of the Governing Equations for the Streaming Potential Problem (Case 1). Boundary

Poisson Equation

Nernst–Planck Equations Axial symmetry Zero normal flux Zero normal flux Zero normal flux

Axial symmetry Slip No slip No slip

AP

Axial symmetry Symmetry1 Potential = 0 Constant potential ( = c , specified) Potential = 0

FQ

Symmetry

Bulk ion conc. (Np = Nn = 1) Bulk ion conc. (Np = Nn = 1)

Normal pressure, P0 (specified) Normal pressure (P = 0)

PQ AB, EF BC, DE CD

1

Navier–Stokes & Continuity Eqs.

Symmetry refers to the condition when no charge is developed at the boundary (n · ∇ = 0).

¯ = 0. For the Nernst–Planck equations, we will use the symmetry condition, n · ∇ ¯ i = 0. Finally, assume no concentration gradient normal to these interfaces, n · ∇N we will use the slip boundary condition for the Navier–Stokes equation. Note that for these boundary conditions to hold, the segments AB and EF should be located sufficiently far away from the capillary entrance. At the inlet plane (AP), we set the concentration of the ions to be equal (Np = Nn = 1) for the Nernst–Planck equations, the potential to zero ( = 0) for the Poisson equation, and the normal pressure to a predetermined value, P0 , calculated from the user-defined axial pressure gradient, for the Navier–Stokes equation. For the evaluation of the streaming potential, the exit reservoir should contain the same electrolyte concentration as the inlet reservoir. Accordingly, we set the conditions at the exit boundary (FQ) as follows. For the Poisson equation, we use ¯ =0 n · ∇

(14.39)

while for the Nernst–Planck equations, we set the concentrations of ions to be equal to the inlet reservoir concentrations, Np = Nn = 1

(14.40)

For the Navier–Stokes equation, we set the normal pressure to zero at this boundary. The boundary conditions on BC, CD, and DE pertaining to the Nernst–Planck equations are those of no net normal flux of ions (no ion penetration) n · j∗∗ i =0

(14.41)

th where j∗∗ i is the number flux of the i ionic species. For the Navier–Stokes equation, the appropriate condition on these boundaries is that of no slip (u = 0). The boundary condition on the capillary wall (CD) for the Poisson equation is set as a constant surface

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potential condition ψ = ψc

or  = c

(14.42)

On the walls BC and DE, the potential is set to zero. The streaming potential problem requires specification of the scaled pressure, P0 , at the channel inlet (AP). The specification of the pressure requires some elaboration. In the subsequent discussion, we will generally refer to an applied axial pressure gradient, px , which is defined as px = −

∂p ∂x

(14.43)

In all the simulations, we fix the pressure at the exit plane FQ as 0. The axial pressure gradient used in these simulations is chosen as 107 Pa/m. Assuming this pressure gradient to be constant between the inlet plane AP and the exit plane FQ, we can write p0 − 0 (14.44) px = 107 = L where p0 is the pressure at plane AP, and L is the overall length of the channel (PQ). From the above relationship, Eq. (14.44), we obtain the pressure at the inlet section p0 = px L = px

L¯ κ

(14.45)

where the scaled length L¯ = 100. The scaled form of this pressure, P0 , is specified as a boundary condition for the Navier–Stokes equation at the inlet section (AP) in Table 14.4. Note that the use of the linear pressure gradient to specify a pressure at the channel inlet section AP is completely arbitrary. As we will observe shortly, the actual axial pressure gradient will vary at different locations in the simulations. The procedure simply allows one to specify a realistic pressure drop across the capillary and a means to provide an appropriate value of P0 at the inlet section. 14.3.3.2 Comparison of Numerical and Analytical Results Solving the steady-state governing equations with the above set of boundary conditions will lead to the prediction of the streaming potential developed between the two reservoirs connected by the capillary microchannel. The parameters that need to be specified are the capillary radius, the bulk electrolyte concentration, the surface potential at the capillary wall, and the pressure at the reservoir inlet plane (AP). For the present simulations, the capillary wall surface potential was set to c = −1. The accuracy of the numerical results can be compared against the analytical results of electroosmotic flow in capillary microchannels described in Chapter 8. The analytical approach described in Chapter 8 employs several assumptions. These include application of linearized Poisson–Boltzmann equation, neglecting axial concentration gradients, and ignoring the entrance/exit effects. In the numerical model, none of these assumptions were made. Consequently, comparison of the numerical and

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analytical results is not straightforward. To perform an accurate comparison of the numerical and analytical results, we first need to assess the behavior of some of the key variables along the capillary as obtained from the numerical solution. Figure 14.12(a) depicts the variations of two important parameters, namely, the electric potential and the pressure along the axis of the cylindrical capillary obtained from the steady-state numerical calculations. The corresponding variations of the electric field component along the axial direction (along x), ¯ as well as the axial pressure gradient are shown in Figure 14.12(b). It is evident that the electric potential and the pressure are virtually constant in the two reservoirs flanking the capillary. Within the capillary, the electric potential shows a non-linear increase with axial position, while the pressure shows a nearly linear variation with axial position. Figure 14.12(b) clearly shows that the axial component of the electric field varies perceptibly along the length of the capillary. The axial pressure gradient within the capillary, on the other hand, shows a more modest variation with axial location. All these quantities undergo marked variations near the capillary entrance and exit regions. It is therefore clearly discernable that a comparison of the results obtained from the detailed numerical calculations with the approximate analytical results is somewhat convoluted. Nevertheless, if we focus our attention to the mid-section of the capillary (corresponding to x¯ = 50), which is sufficiently removed from the channel entrance and exit regions, it is likely that the local conditions at this section from our numerical results will closely emulate those existing in an infinitely long capillary. Furthermore, if the capillary wall surface potential is small, one might expect that the linearization of the Poisson equation in the analytical model will be reasonably accurate. Therefore, we first compare the axial velocity profile obtained from the numerical simulation at the capillary mid-section, x¯ = 50, with the corresponding analytical expression of the velocity profile, Eq. (8.108), which is given by ux (r) ≡ ux =

   r 2  ǫψ  Io (κr) a 2 px c − Ex 1− 1− 4µ a µ Io (κa)

(14.46)

Here, ψc is the surface potential of the capillary wall. Note that to compare the numerical and the analytical velocity profiles, one will need to employ values of px = −∂p/∂x and Ex in Eq. (14.46) that are identical to the values of these parameters at the mid-plane of the capillary (x¯ = 50) obtained from the numerical solution (Figure 14.12). To perform the comparison of the velocity profiles, we conducted the numerical simulation for κa = 1, 5, and 9 using a constant surface potential of c = zeψc /kB T = −1 at the capillary wall. The numerical simulations were conducted with an axial pressure gradient of px = 107 Pa/m between the reservoirs.2 From the steady-state numerical solution of the governing equations, the values of px and Ex at x¯ = 50 and r¯ = 0 were determined. These values were then substituted in Eq. (14.46) to calculate the analytical velocity profile. The comparison of the numerical and 2

This is, however, not the actual pressure gradient at the capillary mid-section, as is evident from Figure 14.12.

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NUMERICAL SIMULATION OF ELECTROKINETIC PHENOMENA

Figure 14.12. (a) Variation of the scaled electric potential and pressure and (b) the correspond¯ and axial pressure gradient (∂P /∂ x) ing dimensionless electric field (E) ¯ along the axis of the capillary and the inlet and outlet reservoirs obtained from the steady-state numerical solution of the governing electrochemical transport equations. Solid lines represent the electric potential and field, while the dashed lines represent the pressure and pressure gradient. The simulations were performed for a scaled surface potential of c = −1 on the capillary wall, an applied pressure gradient of 107 Pa/m between the reservoirs, and a scaled capillary radius of κa = 5. The vertical dotted lines in each figure represent the entrance and exit planes of the capillary microchannel.

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Figure 14.13. Comparison of the steady-state axial velocity profiles at the capillary midsection obtained from the numerical solution with the corresponding predictions of the analytical expression, Eq. (14.46). Symbols are numerical predictions, while lines represent analytical results. The numerical and analytical velocities were compared using the local pressure and electric potential gradients existing at the capillary mid-section, x¯ = 50, obtained from the numerical solution.

analytical velocity profiles is shown in Figure 14.13. It is evident from the figure that the velocity profile obtained numerically at the mid-section of the channel (x¯ = 50) is in remarkably good agreement with the velocity profile obtained for an infinitely long cylindrical capillary using the analytical procedure. It is important to note that the comparison of the numerical and analytical velocity profiles were performed using the values of the local axial pressure gradient and electric field that exists at the mid-section of the capillary (at r¯ = 0 and x¯ = 50). The axial pressure gradient at this position is different from the overall pressure gradient between the reservoirs employed to obtain the numerical solution. Furthermore, the electric field is different at different locations in the capillary. In this context, it now becomes pertinent to verify whether the numerical prediction of the streaming potential, as depicted in Figure 14.12(a), compares with the corresponding analytical prediction of the streaming potential. The numerical prediction of the streaming potential is simply the potential difference observed between the inlet and outlet reservoirs in Figure 14.12(a). The analytical results of Chapter 8 do not directly yield a potential difference between the two reservoirs connected by the capillary. Instead, it provides the following relationship for streaming potential under zero current condition (cf., Eq. 8.143)     Ex ǫψc 2I1 (κa) = 1− f (κa, β, Fcc ) (14.47) px I =0 µσ ∞ (κa)I0 (κa)

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where β = ǫ 2 ψc2 κ 2 /µσ ∞ , and f (κa, β, Fcc ) =

1    Fcc 1 − β 1 − 2A1 /(κa) − A21 /Fcc

Here, Fcc is a correction term for the conduction current, and A1 = I1 (κa)/I0 (κa). Recall the earlier discussion in Chapter 8 that the function f (κa, β, Fcc ) has a sizeable contribution to the streaming potential only for large values of ψc . Equation (14.47) only provides the ratio of the electric field and the pressure gradient that should exist within the capillary at steady state for the streaming potential condition. It is therefore instructive to explore how the potential difference obtained across the capillary in the numerical calculations as shown in Figure 14.12(a) relates to Eq. (14.47). To calculate a potential difference over a given length of the capillary using the analytical approach, the right hand side of Eq. (14.47) was evaluated employing identical capillary radius and other conditions applied to our numerical simulations. In the analytical calculations, the value of Fcc was set to 1.0. From the numerical simulation, the pressure difference, p, over an axial distance L between a location in the inlet reservoir and another location in the outlet reservoir was computed. These locations were selected such that they were equidistant from the capillary mid-section. Dividing the pressure difference by L provided the axial pressure gradient, px (= p/L). Using the pressure gradient in Eq. (14.47), the axial component of the electric field, Ex , was determined. This value of the electric field was then employed to calculate the potential difference over the length L using ψ = Ex L The potential difference thus calculated using the analytical expression, Eq. (14.47), was compared with the potential difference between identical locations in the two reservoirs obtained from the numerical simulations. Figure 14.14 shows the comparison of the two potential differences for different values of scaled capillary radius κa. The comparison is based on the parameters evaluated along the axis of the capillary. Under these conditions, it is evident from Figure 14.14 that the numerically and analytically evaluated potential differences across the capillary are in remarkably good agreement. Furthermore, it is interesting to note that the choice of L does not affect the predicted streaming potential, as long as the two locations in the inlet and outlet reservoirs are equidistant from the capillary mid-plane (x¯ = 50). The numerical results slightly overpredict the potential difference for lower values of κa < 5. However, for κa ≥ 5 the two predictions are virtually identical. The excellent agreement between the numerical and analytical results not only serves as a validation of the numerical model, but also provides considerable insight regarding the development of streaming potential across charged capillaries. First, the comparison of the two solutions were performed on the basis of conditions, particularly the axial pressure gradient and electric field, determined at the mid-point of the capillary (¯r = 0, x¯ = 50). Secondly, the streaming potential in the analytical solution was based solely on the streaming and conduction current, whereas in the numerical

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Figure 14.14. Comparison of the numerical and analytical predictions of the streaming potential for different values of scaled capillary radius.

calculations, it was obtained considering all three modes of ion transport (convection, diffusion, and migration). These comparisons shed some insight regarding how the streaming potential from the analytical solution can be interpreted for a more realistic geometry of a finite length capillary connecting two reservoirs. It is important to note that the streaming potential is measured between the two reservoirs connected by the capillary, and not within the capillary.

14.3.4 Case 2: Transient Analysis of Electrolyte Transport in a Capillary Microchannel We turn our attention now to a slightly different problem, which involves transport of an electrolyte solution through a capillary microchannel representing a membrane pore. In this case, unlike the streaming potential simulations, the ion concentrations in the outlet reservoir at steady-state are not known a priori. Furthermore, instead of a steady-state solution, we solve the time dependent governing equations to simultaneously observe how the steady-state electric potential and the ion concentration distributions are established upon imposition of an axial pressure gradient. The analysis of the transient development of the streaming potential condition across a finite length capillary was recently addressed by Mansouri et al. (2005), who used a similar numerical approach and a computational geometry as described here. They studied the transient distributions of the electric potential, field, and ion concentrations in the system, and assessed the different time scales required for the development of the steady-state potential and the ion concentration fields in the system. 14.3.4.1 Boundary and Initial Conditions We conduct the simulations for a capillary radius of κa = 5, and a constant surface charge density on the capillary wall.

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All geometrical and parametric specifications are identical to the previous case study. The only difference in the present model pertains to specification of an alternate set of boundary conditions that emulate the membrane transport process. The boundary and initial conditions for this transient problem are stated in Table 14.5. For this problem, we demonstrate the use of constant surface charge density boundary conditions on the charged walls of the capillary, as well as the two reservoirs (BC, CD, and DE). Accordingly, on these boundaries, we use the condition (14.48)

−n · [ ǫ∇ψ ] = qc

where qc (C/m2 ) is the surface charge density on the walls. This quantity is provided as an input to the problem. The non-dimensional form of the surface charge density is σc =

ze qc ǫkB T κ

Here we intend to emulate the transport through a membrane pore, and hence, cannot a priori provide the electrolyte concentration in the exit reservoir. Consequently, we specify the convective flux condition at FQ for the Nernst–Planck equations, which is written as j∗∗ i = ni u

(14.49)

TABLE 14.5. Boundary and Initial Conditions of the Governing Equations for Transient Electrokinetic Flow in a Capillary Microchannel (Case 2). Boundary

Poisson Equation

Nernst–Planck Equations Axial symmetry Zero normal flux Zero normal flux

Axial symmetry Slip/symmetry No slip

Zero normal flux

No slip

AP

Axial symmetry Symmetry Constant charge density, qc (specified) Constant charge density, qc (specified) Zero potential

FQ

Symmetry

Bulk ion conc. (Np = Nn = 1) Convective flux ¯ i = 0) (∇N

Normal pressure, P0 (specified) Normal pressure (P = 0)

PQ AB, EF BC, DE

CD

Initial condition: AP

—

—

Navier–Stokes & Continuity Equations

Normal pressure P = 0

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In other words, the diffusional or migration contributions to the ionic flux are assumed to be zero, which implies ¯ i =0 (14.50) ∇N and requires setting the boundary condition for the Poisson equation as ¯ =0 n · ∇

(14.51)

In this case, the reservoir at the capillary outlet will have a lower electrolyte concentration than the feed side reservoir. Although the individual ion concentrations in the outlet reservoir will be lower than the feed reservoir, the electrolyte solution will be electroneutral in this outlet reservoir. All other boundary conditions for the present model are identical to that of the streaming potential simulation case discussed previously. Finally, one needs to provide the initial conditions (at t = 0) for this transient problem. To formulate the initial conditions for the problem, we consider an initial state where the electrolyte solution is in equilibrium with no externally imposed pressure gradient. In this case, the ion distributions are set up in response to the charged walls of the capillary and reservoirs according to Boltzmann distribution. Thus, initially, we calculate the electrolyte concentration in the computational domain from the Poisson and Nernst–Planck equations in absence of any convection (u = 0). Under no-flow conditions, these equations simply become the Poisson–Boltzmann equation. This equation can be solved subject to the conditions of a specified bulk electrolyte concentration at the inflow and exit boundaries (AP and FQ, respectively), and a specified potential on the solid walls (in these simulations, c = −1). The no flow situation is modeled by setting the pressure to zero at AP (P0 = 0). The ion concentration and electric potential distributions thus obtained can serve as the initial conditions for the transient problem. Once the stationary potential distribution is calculated, one can extract the scaled surface charge density, σc , at the capillary and reservoir walls from this solution. This scaled surface charge density is then used in the transient simulations as a boundary condition. The scaled charge density used in the transient calculations was σc = −1.04. The transient problem tracks the modification of the initial stationary distributions of ion concentrations and electric potential due to the fluid motion (convection) once a pressure gradient is applied across the capillary microchannel. The transient problem is solved according to the flowchart shown in Figure 14.10. 14.3.4.2 Transient Simulation Results Figure 14.15 depicts the development of the electric potential and the ion concentration distributions along the capillary axis obtained from the transient numerical simulations immediately after imposing an axial pressure gradient on the system. In Figure 14.15, the axial positions 25 ≤ κx ≤ 75 correspond to the capillary, while the regions κx < 25 and κx > 75 represent the inlet and outlet reservoirs, respectively. The parameter τ = κ 2 Dt corresponding to each solid line represents the scaled time (see Table 14.3) at which the profile was

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NUMERICAL SIMULATION OF ELECTROKINETIC PHENOMENA

Figure 14.15. Transient development of scaled electric potential, , and the positive (Np ) and negative (Nn ) ion concentration profiles along the capillary axis. The scaled capillary radius is κa = 5. The figures on the left hand column show the developing potential and ion concentration profiles immediately after application of the pressure gradient (scaled time, τ varying from 0 to 10), while the right hand side column depicts the corresponding profiles at the later stages of the transient simulations (τ = 50 to steady-state, denoted by SS) (Mansouri et al., 2005).

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determined. The dashed lines marked “SS” indicate the steady-state (τ → ∞) results. The graphs on the left hand column depict the development of the potential and ion concentration profiles during the initial period (τ ≤ 10) after the pressure gradient is applied. The graphs on the right hand column show the corresponding profiles at later stages of the transient simulation (τ ≥ 50). For the simulation parameters used in this study, a scaled time of τ = 1 corresponds to a real time of t = 10 microseconds. The remarkable feature of these simulations is the rapid buildup of the potential difference between the two reservoirs after imposition of the pressure gradient. Within τ = 10, the potential distribution attains values within 5% of the steady-state potential distribution. In contrast, the ion concentration distributions are not affected during these initial times. Both the positive and negative ion distributions remain unchanged from the initial Poisson–Boltzmann distribution during this period. At later stages of the transient simulation, the potential distribution does not undergo any significant change. The ion concentration distributions, however, undergo substantial modification during this long term relaxation of the system toward a steady-state. In the above simulations, the convective flux boundary condition was used for the Nernst–Planck equations at the exit plane (FQ). Accordingly, the ion concentration profiles obtained at longer durations show a distinct difference between the ion concentrations in the inlet and outlet reservoirs. At steady state, the concentrations of both the co- and counterions (Nn and Np , respectively) in the outlet reservoir are lower than the corresponding concentrations in the inlet reservoir. This difference is attributed to the ion rejection by the capillary. The interesting feature of the results is that both the co- and counterions have the same concentration in the outlet reservoir, indicating that this reservoir contains an electroneutral electrolyte solution. Thus, although a charged capillary will reject co-ions, the overall coupled transport of the ions through the capillary self-consistently ensures that the permeate solution is electroneutral, and there is no current transport through the capillary at steady-state. It is evident from these simulations that the steady-state electric potential and ion concentration distributions evolve over largely disparate time scales. The development of the steady-state potential is almost instantaneous, while the equilibrium ion concentrations are established over a much longer time period. For more detailed discussions on these transient simulations, one may refer to Mansouri et al. (2005). 14.3.5 Case 3: Electroosmotic Flow due to Axial Pressure and Electric Potential Gradients We now focus our attention to the flow of an electrolyte solution through a capillary microchannel engendered by the combined influence of axial pressure and electric potential gradients. This general case of electroosmotic flow will be treated as a variant of the steady-state problem discussed in conjunction with the prediction of streaming potential (Case 1). In the present problem, we will simply demonstrate how application of an external electric field at the exit plane of the system in addition to an applied axial pressure gradient modifies the fluid velocity distribution within the capillary microchannel.

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14.3.5.1 Boundary Conditions Since this case involves simultaneous application of a pressure and an electric potential gradient, one can retain the pressure boundary conditions on segments AP and FQ used in Case 1. To consider the applied electric field, one can either apply an electric potential, ψ0 , at the exit plane FQ, or specify the electric field, E0 , at this plane. Here, we demonstrate how the electric field is imposed at the exit plane through the boundary condition ¯ = −E¯ 0 n · ∇

(14.52)

on FQ. Here E¯ 0 is the scaled external electric field, provided as an input to the problem. This boundary condition is specified in the FEMLab program using the displacement vector, D. In terms of the non-dimensionalization used in developing the model, one ¯ = E. ¯ In non-dimensional form, the displacement boundary condition at can write D FQ is specified as ¯ ¯ ¯ D| (14.53) FQ = E|FQ = E0 ix¯ where E¯ 0 is the specified scaled axial electric field. The radial component of the displacement vector is specified as zero at the boundary FQ. The scaled axial electric field is related to the external electric field specified on the boundary FQ, E0 , (V/m) through ze E¯ 0 = E0 kB T κ In terms of the parameters used in the present model, for an electrolyte concentration of c∞ = 10−5 M, when κ = 1.038 × 107 m−1 , a scaled axial field of 0.01 will correspond to an applied electric field of 2.67 × 103 V/m. The electric potential on plane AP is set to zero. On the charged walls of the capillary and the reservoirs (BC, CD, and DE), we employ the constant surface charge density condition, where the charge density is specified as a problem input. All other boundary conditions for this problem are identical to those specified in context of Case 1. Table 14.6 describes these boundary conditions. For the problem, the required input parameters are the applied pressure at the inlet (AP), the electric field at the exit plane (FQ), and the charge densities on the walls (BC, CD, and DE). As in case 2, the scaled charge density at the solid walls (BC, CD, DE) was set to σc = −1.04, which roughly corresponds to a scaled surface potential of c = −1 in absence of any flow. 14.3.5.2 Velocity Profiles in Electroosmotic Flow When discussing electrokinetic flow through a capillary microchannel in Chapter 8, it was shown within the context of the linearized theory (based on the linearized Poisson–Boltzmann equation) that the overall electrokinetic flow in presence of both axial pressure and electric potential gradients is a simple superposition of purely pressure driven flow and purely electroosmotic flow. Here, we first test whether such an assumption is valid when the complete problem formulation is employed. The complete problem formulation in the present numerical simulation does not involve linearization of the governing equations, and considers a finite length capillary. Furthermore, with a constant surface

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TABLE 14.6. Boundary Conditions of the Governing Equations for Electroosmotic Flow in Presence of Combined Axial Pressure and Electric Potential Gradients (Case 3). Boundary PQ AB, EF BC, DE

CD

AP FQ

Poisson Equation

Nernst–Planck Equations

Axial symmetry Symmetry Constant charge density, qc (specified) Constant charge density, qc (specified) Potential = 0

Axial symmetry Zero normal flux Zero normal flux

Axial symmetry Slip/symmetry No slip

Zero normal flux

No slip

Bulk ion conc. (Np = Nn = 1) Bulk ion conc. (Np = Nn = 1)

Normal pressure, P0 (specified) Normal pressure (P = 0)

Electric field, E¯ 0 (specified)

Navier–Stokes & Continuity Eqs.

charge density boundary condition, the surface potential on the capillary wall can vary axially. Figure 14.16 depicts the scaled axial velocity profile, ux /(κD), at the mid-section of the capillary microchannel obtained under different combinations of the axial pressure gradient, px = −∂p/∂x, and scaled axial electric field, E¯ 0 , applied at the plane FQ. The velocity profile shown by the dashed line for purely pressure driven flow (with px = 107 Pa/m) was obtained by numerically solving the Navier–Stokes equation for an uncharged capillary. No electrokinetic effects were considered in obtaining this velocity distribution. The velocity profile for a purely electroosmotic flow (px = 0 and E¯ 0 = 0.01) was obtained by solving the complete electrokinetic transport model numerically, after setting the axial pressure gradient to zero (using P = 0 at AP). In this case, the electric field was directed along the the positive x-coordinate. For the purely pressure driven flow, the velocity profile is parabolic, while for the electroosmotic flow, the velocity distribution has a flatter profile near the channel axis. The symbols in Figure 14.16 were obtained by adding the two above results for purely pressure driven and electroosmotic flows. Finally, Figure 14.16 also shows the velocity profile when both the axial pressure gradient and the electric field are simultaneously applied to the electrolyte solution, as shown by the solid line (with px = 107 Pa/m and E¯ 0 = 0.01). The resulting velocity profile obtained from the numerical solution (solid line) has a slight deviation from the corresponding velocity distribution obtained by linear superposition of the pressure driven and electroosmotic velocity distributions (symbols). The maximum deviation between these two results is about 2%. While this difference definitely indicates that the linear superposition of the pressure driven and electroosmotic velocities is not exact, it is also evident that such a superposition is a fairly good approximation. The slight difference between the two results originates from the fact that the local electric fields and pressure gradients at the capillary mid-section are quite different in the case of purely pressure driven flow, purely electroosmotic flow, and electrokinetic

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Figure 14.16. Variation of the steady-state axial velocity, ux /(κD), with scaled radial position at the capillary mid-section (x¯ = 50). Solid line: numerical simulation of electrokinetic flow under the combined influence of pressure (px = 107 Pa/m) and an applied electric field (E¯ 0 = 0.01). Dashed line: purely pressure driven flow (no electrokinetic effect, E¯ 0 = 0). Dashdotted line: purely electroosmotic flow (E¯ 0 = 0.01, px = 0). Symbols: summation of purely pressure driven and electroosmotic flows (dashed and dash-dotted lines). The scaled capillary radius is 5, and the scaled capillary wall surface charge density is σc = −1.04.

flow under the combined influence of pressure and electric field. To demonstrate this, we plot the axial electric fields and the axial pressure gradients along the capillary axis in Figure 14.17 corresponding to the different flow situations depicted in Figure 14.16. For a purely pressure driven flow in absence of electrokinetic effects, the electric field is zero. The scaled axial electric fields obtained for the purely electroosmotic flow (dash-dotted line) and the combined pressure and electric field driven electrokinetic flow (solid line) are slightly different at the mid-section of the capillary, as seen from Figure 14.17(a). It is also important to note that the field inside the capillary microchannel is substantially different from the applied field, E¯ 0 = 0.01. The axial pressure gradients at the capillary mid-section corresponding to purely pressure driven flow (dashed line), purely electroosmotic flow (dash-dotted line), and the combined electrokinetic flow (solid line) are quite different, as is evident from Figure 14.17(b). One may note that there exists an adverse axial pressure gradient within the capillary (dash-dotted line) during purely electroosmotic flow. For the purely pressure driven flow, the axial pressure gradient within the capillary is constant. It is discernable from Figure 14.17 that the three flow scenarios considered here result in different axial electric fields and axial pressure gradients at the channel mid-section. In this context, it is indeed remarkable that despite these different driving forces, the results from superposition of pressure driven and electroosmotic velocities are in such close proximity to the complete electrokinetic velocity profile in presence of combined axial pressure gradient and electric field.

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Figure 14.17. Axial variation of the electric field, E¯ x , and the pressure gradient, ∂P /∂ x, ¯ along the capillary axis (¯r = 0) corresponding to the results shown in Figure 14.16. The vertical dashed lines represent the capillary entrance, mid-section, and the exit. The different line types correspond to the line types shown in Figure 14.16.

The simulation results for the axial velocity profile in the capillary under the influence of oppositely directed pressure and electric field gradients are presented next. The simulations were performed by imposing external electric fields of different magnitudes acting along the capillary axis and observing the steady-state velocity profile of the electrolyte at the channel mid-section. The scaled electric field, denoted by E¯ 0 , and applied at the plane FQ, was varied from zero to −0.05. When the external field is zero, we have the situation of pressure driven electrokinetic flow, with the pressure gradient directed along the positive x coordinate. This velocity profile is depicted by the topmost curve of Figure 14.18. Note that the velocity profile obtained for this case will be slightly different from the purely pressure driven velocity distribution shown in Figure 14.16. This is owing to the fact that in the present case, which considers the complete electrokinetic transport, the electric field generated by the streaming potential will counter the pressure driving force and slightly diminish the velocity.

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NUMERICAL SIMULATION OF ELECTROKINETIC PHENOMENA

Figure 14.18. Variation of the steady-state axial velocity with radial position at the capillary mid-section under the combined influence of oppositely directed pressure gradient and electric field (E¯ 0 ). The axial pressure gradient used in all cases was px = 107 Pa/m. The scaled capillary radius is 5, and the scaled capillary wall surface charge density is σc = −1.04.

The electric field, E¯ 0 , is now applied in a direction opposite to the axial pressure gradient (along negative x direction). The ensuing steady-state velocity profiles are depicted in Figure 14.18. Positive values of the scaled axial velocity represent the flow along the positive axial direction (along the direction of the pressure gradient, px ), while negative values indicate a reversal of the flow. The simulations show that the parabolic velocity profile observed under the influence of the pressure gradient is modified upon application of a counteracting externally imposed electric field. For small electric fields, the velocity decreases slightly from the purely pressure driven flow. The velocity near the capillary wall reverses direction upon application of the electric field. For the scaled field of −0.01, we observe that the fluid near the capillary wall flows in a direction opposite to the flow in the core of the capillary microchannel. In this case, the velocity is zero at two radial locations, namely, at the capillary wall (no-slip), and at another location (¯r ≃ 2) between the capillary axis and the wall. This second location is the stationary plane where the fluid does not undergo any axial motion. Between this stationary plane and the capillary wall, there exists a location (corresponding to the maximum negative velocity), where the shear stresses will vanish. Further increasing the electric field (E¯ 0 = −0.05) completely reverses the flow direction. The different types of simulation results presented above indicate the general applicability of the numerical simulation procedure to address a wide range of problems associated with electrokinetic transport of an electrolyte through a charged capillary microchannel. The advantage of such a numerical simulation code is the flexibility of modifying the geometry, surface properties, and boundary conditions, leading to different applications. Although a symmetric electrolyte with identical diffusivities of different ionic species was considered in the foregoing

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ANALYSIS OF ELECTROPHORETIC MOBILITY

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simulations, it is fairly straightforward to incorporate different diffusivities of different ionic species and asymmetric electrolytes in the current model. Furthermore, multi-component electrolytes (with more than two ionic species) can be considered through inclusion of additional Nernst–Planck equations in the model. Although the initial development and validation of a numerical methodology can be somewhat tedious, a sufficiently general numerical code can eventually perform simulation tasks for more realistic situations compared to what can be addressed through approximate analytic methods of solving the governing electrokinetic transport equations.

14.4

ANALYSIS OF ELECTROPHORETIC MOBILITY

The electrophoretic mobility of a single spherical particle in a large reservoir containing an electrolyte solution was discussed extensively in Chapter 9. The classical approach of Henry (1931) was employed to determine the mobility for low surface (zeta) potentials on the particle. Furthermore, it was stated that rigorous solutions accounting for the convective relaxation effects, such as the perturbation solution of O’Brien and White (1978), leads to a deviation of the electrophoretic mobility from the values predicted by Henry’s solution at higher values of particle surface potential. Of particular interest in this context is the fact that for large values of the scaled particle size, κap > 3, the plot of electrophoretic mobility, η, versus zeta potential attains a maximum between 4 < |zeζp /kB T | < 7, where ζp is the particle zeta potential. In this section, we aim to reproduce these results for selected values of the parameters κap and the scaled particle surface potential, zeζp /kB T , using a finite element solution of the governing electrochemical transport equations representing the electrophoresis of a spherical particle of radius ap . As discussed in Chapter 9, the general governing equations for electrophoresis are the Navier–Stokes and continuity equations, the Poisson equation, and the Nernst– Planck equations. These equations need to be solved simultaneously for a charged particle moving under the influence of an applied electric field in an electrolyte solution, which is stationary at an infinite distance from the particle. Direct numerical simulations of electrophoretic mobility of a single spherical particle in an unbounded fluid are relatively scarce. The perturbation techniques of Wiersema et al. (1966) and O’Brien and White (1978) are still considered to be the most robust methodologies for obtaining a solution of the problem. Several numerical methodologies for calculation of the electrophoretic mobility have subsequently been developed, some of which essentially follow the approach of Teubner (1982). Teubner’s analysis starts from the observation that amongst the equations governing electrophoresis, namely, the Stokes and continuity equations (incorporating electrical body forces) for fluid velocity, the Poisson equation, and the Nernst–Planck equations, only the equations governing fluid velocity are linear. The Nernst–Planck equations remain non-linear even for small applied electric fields. He then proceeded to develop an elegant methodology for circumventing the difficulties associated with the nonlinearities in the Nernst–Planck equations. Based on this methodology, Shugai et al.

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(1997) and Shugai and Carnie (1999) numerically calculated the electrophoretic mobilities of two spherical particles, and for spherical particles in bounded flows. Previously, Keh and Chen (1989a,b) also computed the mobilities for two particles using a bispherical coordinate system. More recently, numerical simulations of electrophoresis of charged particles in a straight cylindrical capillary were reported by Hsu and Hung (2003), as well as Hsu et al. (2004). A direct simulation of electrophoresis of two particles in a rectangular microchannel has also been reported by Ye and Li (2004). Although most of the recent numerical studies mentioned above appear to address electrophoresis in complex geometries, such as, particle motion in bounded flows, or multiparticle systems, there are several simplifications inherent in many of the studies. Most of these simplifications are related to avoiding direct solution of the Nernst–Planck equations governing the ionic movement in the electrolyte solution. The approach of O’Brien and White (1978) is based on perturbation analysis. The methodologies based on Teubner’s analysis (Shugai et al., 1997; Shugai and Carnie, 1999) essentially rely on circumventing the direct solution of the Nernst–Planck equations. Many numerical solutions are based on the assumption of Boltzmann distribution of ions (Hsu and Hung, 2003; Hsu et al., 2004). These techniques are similar in scope as the method of Henry (1931), which neglects convective relaxation effects altogether. Finally, there are approaches which do not explicitly consider the electric double layer in the analysis of the motion, but employ a slip velocity at an arbitrary slip plane surrounding the particle to solve the Navier–Stokes equations for fluid flow (in absence of electrical body forces) (Ye and Li, 2004). The governing equations for electrophoresis might seem fairly tractable, particularly after considering the multiphysics formulation of electroosmosis in the previous section. However, two factors complicate the problem dramatically. The first is associated with the severe non-linearity of the Nernst–Planck equations. As was evident from the foregoing discussions, most numerical treatments of electrophoresis tend to bypass the direct solution of these equations. The second issue is related to properly interpreting the force balance on the particle that leads to the steady-state translational velocity of the particle in an applied field. Anderson (1989) discussed this matter elegantly. Consider the scenario depicted in Figure 14.19. When a negatively charged particle is immersed in an electrolyte solution, the counterions form a positively charged double layer sheath around the particle. Combined, the particle and the sheath form an electroneutral body. The question then is, why will this electroneutral object (particle plus the oppositely charged sheath) move under the influence of an applied electric field? The answer to this question lies in the fact that the positively charged double layer sheath is mobile. In other words, the ionic sheath surrounding the particle can move under the influence of the external electric field. The resulting electroosmotic fluid flow occurs in a direction that is opposite to the direction of the electrical force acting on the particle. The velocity field in this sheath determines the velocity in the fluid outside the charged sheath (the external electroneutral fluid). Consequently, the outer boundary of the charged sheath (denoted as the slip plane) will have a velocity that differs from the actual particle velocity. Thus, if one can calculate the velocity

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589

Figure 14.19. Schematic representation of electrophoresis of a charged spherical particle (shown in gray) in an electrolyte solution under the influence of a uniform external electric field, E∞ . The negatively charged particle and the positively charged electric double layer sheath surrounding it (dashed circle) is electrically neutral. The fluid outside the dashed line denoted as the slip plane is also electrically neutral.

at the slip plane relative to a stationary particle, the electrophoretic mobility of the particle can be inferred from this slip plane velocity. The above picture is considerably different from the case of electrophoretic motion of a charged particle immersed in a charge free dielectric liquid. Here, the fluid has no free charge (ions) to “neutralize” the charge of the particle. Furthermore, due to absence of free charge, there is no electric double layer sheath or an electroosmotic flow. Hence, the steady-state particle velocity in this case arises simply from the electrical force acting directly on the charged particle due to the external field and the counteracting fluid drag force. Note that this fluid drag force is simply the force that even an uncharged particle moving through the viscous fluid will experience. In context of Figure 14.19, for such types of motion, the slip plane moves to infinity. For any finite electrolyte concentration, when a double layer sheath is formed around the charged particle, the net particle motion arises from a combination of three forces. These are, (i) the electrical force acting on the particle due to the imposed field, (ii) the fluid drag force acting on the particle in absence of any charge effects when the particle translates under the influence of the electrical force, and (iii) the excess drag force on the particle due to the counteracting electroosmotic flow (Teubner, 1982; Shugai and Carnie, 1999). The overall steady-state velocity of the particle is governed by a balance of the above three forces. If one is to model the complete transient development of the electrophoretic motion of the particle, a highly sophisticated numerical method is needed, which tracks the movement of the particle in the electric field. Such particle tracking algorithms are becoming increasingly accessible in the field of computational fluid dynamics (Ye and Li, 2004). However, implementation of such an algorithm in a complete electrokinetic model is still somewhat complicated. In particular, the software we

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use for the simulations in this book (FEMLab) does not at present have a robust implementation of a particle tracking algorithm for a multiphysics problem. Typically, finite element methods allow tracking of a moving particle by moving the nodal points of a mesh, a technique akin to the Lagrangian approach. However, in such techniques, the mesh can become considerably distorted (the quality of the elements can become extremely poor) over successive time steps. Thus, one needs to have a methodology for restoring the element quality after several moves of the particle. The implementation of this Lagrangian particle tracking method is fairly straightforward in context of a fluid dynamics problem. However, when one considers the coupled solution of Poisson and Nernst–Planck equations along with the Navier–Stokes equation, the problem becomes even more complicated from a numerical perspective. To avoid the above-mentioned difficulty, we will present here the simulation of the steady-state electrophoretic motion of a particle. The approach we discuss does not require the particle position to be updated. Instead, we fix the coordinate system at the center of the particle, assuming it to be stationary. In this particle fixed reference frame, the fluid surrounding the particle will appear to be moving in a direction opposite to the particle. One can then infer the particle velocity with respect to a stationary observer using the fluid velocity sufficiently far from the particle. The simulations only provide the electroosmotic velocity for finite locations of the slip plane surrounding the particle, and hence, are not applicable directly to obtain the electrophoretic mobility of particles in pure dielectric fluids. Notwithstanding these limitations, the numerical methodology presented here is adequate to provide comparisons of the electrophoretic mobility with some of the analytical and numerical results discussed in Chapter 9. 14.4.1 Problem Formulation Consider a spherical particle of radius ap suspended in a large volume of electrolyte as shown in Figure 14.20(a). Upon application of an electric field, E∞ , the negatively charged particle starts to translate toward the positive electrode. The steady state velocity of the particle, up , is the unknown we wish to determine from the numerical simulation. For the numerical calculations, the geometry is represented in cylindrical coordinates, with the particle center located at the origin (O), Figure 14.20(b). In this particle fixed reference frame, with the electric field aligned along the x axis, we can write the three dimensional problem as an axi-symmetric two dimensional model in cylindrical coordinates (r,x). In the particle fixed reference frame, the fluid flows in the direction opposite to the direction of particle motion. The far-field fluid velocity obtained from the solution of the governing equations is used to calculate the particle electrophoretic mobility. The computational domain used for the simulations is represented by the shaded region in Figure 14.20(b) which is obtained by the intersection of the rectangular domain ABQP and the semi-circle CDE. We will only restrict our attention to solving the electrochemical transport equations in the shaded region (within the electrolyte solution). For a more general solution, one can additionally solve a Laplace equation for the electric potential distribution within the charged particle (inside the domain CDE). However, we will simply assume the particle to have a constant

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591

Figure 14.20. Schematic representation of (a) electrophoresis of a charged spherical particle in an electrolyte solution under the influence of a uniform external electric field, and (b) the axi-symmetric cylindrical computational geometry used for analysis of the electrophoretic mobility. The origin O is set at the center of the particle.

potential or charge density over its surface, and avoid solving the additional Laplace equation within the particle. This is generally justified when the particle dielectric permittivity is much smaller than the dielectric permittivity of the surrounding fluid. The equations governing the problem were described in Section 9.2. Here we briefly reiterate the steady-state forms of these governing equations and pertinent boundary conditions. The electric potential distribution in the electrolyte solution surrounding the particle is given by the Poisson equation  ǫ∇ 2 ψ = −ρf = − zi eni (14.54) i

where ǫ is the dielectric permittivity of the fluid, ρf is the volumetric free charge density, zi and ni are the valency and the number concentration of the i th ionic

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species, respectively. The fluid flow is given by the Navier–Stokes and continuity equations, ρu · ∇u = −∇p + µ∇ 2 u + ρg + ρf E

(14.55)

∇ ·u =0

(14.56)

and

where u is the fluid velocity vector, with the velocity components along the r and x directions represented by u and v, respectively, ρ is the fluid density, µ is the viscosity, g is the gravitational acceleration, and E = −∇ψ is the electric field. Finally, the ion concentration distributions are given in terms of the Nernst–Planck equations,   zi eDi ni ∇ · j∗∗ ∇ψ =0 = ∇ · n u − D ∇n − i i i i kB T

(14.57)

where Di is the diffusivity of the i th ionic species. One now needs to specify the boundary conditions for the problem. Referring back to Figure 14.20(b), one can apply the axial symmetry condition for all the governing equations along the line segment PQ (more specifically, along PC and EQ). We next focus on the boundary represented by the surface of the spherical particle (CDE). The electrical boundary condition for the Poisson equation can be defined as −ǫ n · ∇ψ = qp

(14.58)

for a constant surface charge density, qp , on the particle. The requirement for the electrophoresis problem is that the particle charge density is not modified upon imposition of the external electric field. To obtain the charge density, one can initially solve the Poisson and Nernst–Planck equations in absence of any external field (and any fluid flow) using a constant surface potential at the particle. In this case, the boundary condition on CDE is set as ψ = ζp

(14.59)

where ζp is the specified zeta potential on the particle surface. From the solution of the Poisson and Nernst–Planck equations (with no fluid flow), one can then calculate the surface charge density, qp , on the particle corresponding to the specified surface potential. Following this, during the solution of the complete problem in presence of an external field, one reverts to the constant surface charge boundary condition, Eq. (14.58), where the calculated value of the surface charge density is used. Assuming the spherical particle to be fixed in space, the no slip condition for the Navier–Stokes equation on the boundary CDE becomes u=0

(14.60)

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Finally, for the Nernst–Planck equations, we impose the condition that no electrolyte ions penetrate the sphere surface   zi eni Di Di ∇ni + ∇ψ · n = 0 and u · n = 0 on CDE (14.61) kB T In the above equations, n is a unit outward normal vector (pointing toward the fluid) on the sphere surface. We now turn our attention to the boundary conditions far away from the sphere, on the outer boundaries of the computational domain, denoted by the line segments PA, AB, and BQ. The electrical boundary condition for the Poisson equation on these three boundaries should be (14.62)

∇ψ = −E∞

where E∞ = E∞ ix is the applied electric field acting along the positive x direction. However, implementing this Neumann condition on all the outer boundaries will not yield a solution for the governing equations. To obtain a solution, one needs to specify the electric potential, , at least on one of these external boundaries. Therefore, we implement the electrical boundary condition on these outer boundaries as follows. At the outset, we assume that the electrical potential in absence of the charged particle is zero on the plane x = 0. Now, when a constant external electric field, E∞ , is applied, the potentials on the planes PA and QB can be written as ψ = −xE∞

at PA and QB

where x is the axial coordinate of these planes. Using the above procedure, the boundary conditions on the segments PA and QB are specified as Dirichlet conditions in terms of the electric potentials. On the boundary AB, the imposed electric field is specified as a boundary condition for the electric displacement. In other words, D = ǫE∞ = ǫE∞ ix

on AB

For the Navier–Stokes and continuity equations, the appropriate far field boundary condition is that the overall fluid stress (both pressure and viscous stresses) vanishes. Accordingly, the total fluid stress on the line segment AB is expressed using    = = σ · n = − p I + µ ∇u + (∇u)T · n = 0

(14.63)

ni = ni∞

(14.64)

where p is the hydrostatic pressure. On the segments PA and QB, we assign a zero pressure condition. Finally, for the Nernst–Planck equations, the ion concentrations in the far field (on segments PA, AB, and BQ) are given by their bulk values

 which should satisfy the bulk electroneutrality condition, zi ni∞ = 0. It is important to note that the far field boundary conditions should strictly apply at an infinite distance from the particle. To ensure that application of these boundary

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NUMERICAL SIMULATION OF ELECTROKINETIC PHENOMENA

conditions on the finite computational domain does not introduce appreciable errors in the calculations, one needs to ensure that the radius of the cylindrical domain b is significantly large compared to the particle radius. In all calculations, we consider an outer domain that has a radius equal to 12 to 20 times the radius of the particle. It is also ensured that the boundaries AP and BQ are located at a sufficiently large distance from the particle. Typically, the distances of these boundaries from the particle center were also set at 12 to 20 times the particle radius. Using such a large domain ensures that the fluid velocity at the outer boundaries of the domain are not significantly affected by the particle. One should recall from our discussion of hydrodynamic interactions in Chapter 6 that the hydrodynamic effects of a particle on the Stokes flow in the surrounding fluid extends to very large distances. Consequently, the numerical solutions obtained over a finite domain using the boundary condition given by Eq. (14.63) will never be “exact”. However, noting that all numerical solutions are approximate to within some preset tolerance, it is sufficient to place the outer boundary at a distance where the perturbation of the fluid velocity due to the particle is below the preset numerical tolerance. In the present calculations, setting b = 12ap − 20ap ensures that the hydrodynamic perturbations of the far field velocity due to the particle are adequately suppressed below the numerical accuracy of the finite element program. One can compare the size of the domain used in this problem with the requirement imposed in the solution of the Poisson–Boltzmann equation (Section 14.2), where we used b = 5ap for the location of the outer boundary. Clearly, the computational domain in the present problem is larger due to the long-range nature of hydrodynamic interactions. If one uses a domain that is sufficiently large to attenuate the hydrodynamic interactions, it is imperative that the electrical boundary conditions will also be valid at this large distance, particularly when the parameter κap , where κ is the inverse Debye length, is large. The problem is solved for a symmetric (z : z) electrolyte. Additionally the ionic diffusivities are assumed to be identical for every species as was done in the case of electroosmotic flow discussed in the previous section. All lengths are scaled with respect to the particle radius, ap . Thus, the scaled radial and axial coordinates are represented as r¯ = r/ap and x¯ = x/ap , respectively. This scaling is different from our use of the Debye length, κ −1 , as the length scale in the previous problems discussed in this chapter. The governing equations are non-dimensionalized using the scaled variables described in Table 14.7. The electric field and velocity scalings used here are identical to those used in Chapter 9 in context of the O’Brien and White (1978) model of electrophoretic mobility. The non-dimensional equations are written using the scaled variables as follows. Poisson equation: 1 ∇¯ 2  = − (κap )2 (Np − Nn ) 2

(14.65)

¯ ∗ = −∇P ¯ + ∇¯ 2 u∗ + 1 (κap )2 β ∗ (Np − Nn ) ρu ¯ ∗ · ∇u 2

(14.66)

Navier–Stokes equation:

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595

TABLE 14.7. Scaled Variables Used for Non-Dimensionalizing the Governing Equations for Electrophoresis. Dimensionless Variable

Expression zeψ kB T

Electric potential, 

Particle radius, κap Ion concentration, Ni Gradient operator, ∇¯ Density, ρ¯ Pressure, P

Diffusivity, D¯ i Electric field, β ∗ Particle charge, Q∗p

Velocity, u∗

ap




1 ǫ



2z2 e2 ni∞ ǫkB T ni ni∞ ap ∇   ǫ kB T 2 ρ µ2 ze ap ze kB T

2

p

  µDi ze 2 ǫ kB T   ap ze E kB T   ze Qp ǫkB T ap µap ǫ



ze kB T

2

u

Nernst–Planck equation: ¯ i − sign(zi )D¯ i Ni ∇] ¯ ∇¯ · [u∗ Ni − D¯ i ∇N =0

(14.67)

For the present problem involving a binary electrolyte, we deal with two Nernst– Planck equations, one for the positively charged ion, denoted by subscript p, and a second for the negatively charged ion, represented by subscript n. For each type of ionic species, the scaled Nernst–Planck equation should incorporate the appropriate sign of the ionic charge in the migration flux term. The parameter β ∗ in Eq. (14.66) represents the scaled electric field,   ap ze E (14.68) β∗ = kB T where E is the local electric field.

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One final point needs to be clarified regarding the calculation of the electrophoretic mobility using the above equations. This pertains to a statement made at the outset regarding how the equations are written in a particle fixed reference frame. One should note that to a stationary observer, a particle bearing a total charge Qp , under the influence of an externally imposed electric field of magnitude E∞ , will undergo an electrophoretic motion even when there are no ions present in the fluid surrounding it. The steady state particle velocity in this case will evolve from a balance of the electrical force, Qp E∞ , and the Stokes drag force, 6π µap up . In this so-called Hückel limit (κap = 0), the particle velocity is given by up =

Q p E∞ 6π µap

However, one can observe from Eq. (14.66) that the fluid velocity becomes zero as κap → 0, since there is no electrical body force acting on the fluid in this case. This is because, the present set of equations written in a particle fixed reference frame only provides the electroosmotic flow of the electrolyte surrounding the particle. In other words, as κap → 0, the electrophoretic mobility of the particle predicted by the present model will approach zero, much in the manner in which the cell models for concentrated suspensions predict a zero mobility for finite volume fractions of particles as κap → 0. Thus, one should only employ the present model for finite values of the parameter κap . In this study, we will restrict our attention to the calculation of the mobility for κap > 1. In the following, we briefly discuss numerical models obtained by simplification of the above governing equations that represent Henry’s (1931) approach, as well as the linear perturbation analysis presented in Chapter 9. 14.4.1.1 Formulation of Henry’s Problem The formulation described above represents the general set of governing equations for electrophoresis, without any assumption of linear superposition of the external electric field on the electric field generated by the charged particle. In Chapter 9, when discussing the linear perturbation theory, we noted that the methodology involves two steps, namely, evaluation of the electric potential distribution, ψ eq , around a charged particle, followed by imposing the external field, E∞ , and adding the local potential perturbation, δψ, due to this field on ψ eq . One might implement this methodology by separately solving a Laplace equation governing the potential distribution due to the applied electric field (as in Henry’s approach). In the following, we present this procedure, which numerically emulates Henry’s technique. Recall that Henry’s approach involved solving the Poisson–Boltzmann equation for the potential distribution due to the charged particle, ψ, the Laplace equation for the potential distribution due to the externally imposed electric field, φ, and the Navier–Stokes equation. This approach does not consider convective relaxation effects, and hence, assumes that the ion distributions are not affected by the fluid flow. Consequently, one does not need to explicitly solve the Nernst–Planck equations.

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597

The numerical implementation of Henry’s approach involves solving the following equations, written in scaled form. Poisson–Boltzmann equation: ∇¯ 2  eq = (κap )2 sinh( eq )

(14.69)

where  eq = zeψ eq /kB T is the equilibrium potential distribution around the charged particle. Laplace equation: ∇¯ 2  = 0

(14.70)

where  = zeφ/kB T represents the potential distribution due to the applied electric field. Navier–Stokes equation: ¯ ∗ = −∇P ¯ + ∇¯ 2 u∗ + (κap )2 sinh( eq )∇( ¯ eq + ) ρu ¯ ∗ · ∇u

(14.71)

Here, the electric field used in the Navier–Stokes equation is given in terms of the ¯ eq + ). total potential gradient, ∇( The boundary conditions for the above set of equations are written in a slightly different manner. For the Poisson–Boltzmann equation, we set the electric potential on the particle surface (boundary CDE) to  eq = p = zeζp /kB T . At the outer boundaries of the domain (PA, AB, BQ), we set  eq = 0. For the Laplace equation, as in Henry’s approach, the charge density is set to zero at the particle surface (CDE). On the outer boundaries PA and QB, the electric potential,  is given by ∗  = −xβ ¯ ∞

at PA and QB

(14.72)

∗ is the scaled applied electric field. On the boundary AB, the applied electric where β∞ field is specified as the electric displacement boundary condition for the Laplace equation. The boundary conditions for the Navier–Stokes equation are similar to the general formulation discussed earlier. The above formulation based on Henry’s approximation is considerably simpler than the general procedure for solving the problem. However, the formulation completely ignores any redistribution of ion concentrations around the particle due to the fluid convection.

14.4.1.2 Formulation of the Perturbation Problem This formulation implements the equations described in Section 9.3.1, and is similar to the set of governing equations solved by Wiersema et al. (1966) and O’Brien and White (1978). The pertinent variables are written as sums of the equilibrium value plus the perturbation

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in presence of the applied field. In scaled form, these variables are Potential:  eq + δ Velocity: 0 + u∗ eq

Ion concentration: Ni + δNi Pressure: P eq + δP For the equilibrium situation, we solve Eqs. (9.43) to (9.45), which yields the Poisson–Boltzmann equation as in Eq. (14.69). Solution of this problem provides eq the equilibrium parameters, namely,  eq and the scaled ion concentrations, Ni . Following this, the linear perturbation equations, namely, Eqs. (9.49) to (9.52) are solved. In scaled form, these equations are given by Poisson equation: 1 (14.73) ∇¯ 2 δ = − (κap )2 (δNp − δNn ) 2 Navier–Stokes equation:  1 ¯ ∗ = −∇δP ¯ ¯ ¯ eq u∗ · ∇u + ∇¯ 2 u∗ − (κap )2 (Npeq − Nneq )∇δ + (δNp − δNn )∇ 2 (14.74) Nernst–Planck equation:    eq ¯ i − sign(zi )D¯ i Nieq ∇δ ¯ ¯ eq = 0 ∇¯ · u∗ Ni − D¯ i ∇δN + δNi ∇

(14.75)

where Ni = Np , Nn for the positive and negative ionic species, respectively. The boundary conditions for this perturbation problem are given by Eqs. (9.53) and (9.54). A solution of these equations will provide the fluid velocity field around the particle, from which the electrophoretic mobility can be determined. The perturbation formulation, unlike the Henry formulation, accounts for the convective relaxation of the electric double layer. However, it is applicable when the magnitude of the externally imposed electric field is much smaller than the field due to the charged particle. 14.4.2 Mesh Generation and Numerical Solution As with all the other problems discussed in this chapter, the computational domain of Figure 14.20(b) was discretized using a non-uniform mesh consisting of quadratic triangular elements. A typical mesh used in the numerical calculations is shown in Figure 14.21. The nature of the physical problem imposes serious demands on the mesh. On one hand, to ensure that the far field fluid velocity is not affected by the particle, and that the boundary conditions at the outer periphery of the domain are sufficiently accurate, one needs to consider a large computational domain. Secondly, since the electric potential and ion concentrations decay rapidly near the particle, particularly for larger values of the parameter κap , one needs to have a sufficiently

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599

Figure 14.21. Finite element mesh used for modeling electrophoretic mobility of a charged spherical particle. (a) Discretization of the computational domain into three regions, namely, outer, middle, and innermost. (b) The refined mesh around the particle (innermost region).

refined mesh near the particle surface. Consequently, finite element calculations for electrophoretic mobility can become highly computation intensive. The discretization of the computational domain is performed by selecting three regions surrounding the particle as shown in Figure 14.21(a). The thickness of the innermost spherical region surrounding the particle is approximately 2κ −1 , and it

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contains a highly refined mesh. The decay in the electric potential and ion concentrations are generally confined within this domain. The middle region has an outer radius of approximately 6ap . The fluid velocity virtually attains the far field values at this outer radius. This intermediate shell has a slightly coarser mesh compared to the innermost region. Finally, the outermost region has the coarsest mesh. The flow in this outer region is generally uniform, and no electrical effects due to the particle are perceptible in this outer region. In this context, one may even solve the Navier– Stokes equation in this outer region without explicitly considering the electrical body forces. Furthermore, the solution of the Laplace equation may suffice in this region, since the electrolyte solution is electroneutral. The division of the computational domain into these three sub-domains ensures that the highly non-linear Poisson and Nernst–Planck equations are solved accurately near the particle surface. The interior boundaries demarcating the innermost and middle, as well as, the middle and outer regions are “continuous”, which implies that for every equation, the corresponding continuity condition applies at these internal boundaries. The mesh in these calculations generally consist of approximately 20,000 elements. The maximum element size in the innermost domain is selected such that the decay in the electric potential and ion concentrations around the particle is captured accurately. This refined region is depicted in Figure 14.21(b). The maximum element size, as well as the thickness of this spherical shell domain is dependent on the value of the parameter κap . As κap increases, the domain shrinks in thickness. The numerical solution of the governing equations is performed to obtain the steady state far field velocity of the fluid under an applied electric field. The parameters and their ranges used in the simulations are shown in Table 14.8. The solution methodology involves an initialization step, which requires solving the Poisson and Nernst–Planck equations simultaneously to obtain the ion concentration and electric potential distributions in absence of any externally imposed electric field. For this initialization step, a constant surface potential condition is used on the particle. The particle surface charge density calculated from this initial solution is then substituted as a boundary condition to the Poisson equation for subsequent computations. The Poisson and Nernst–Planck equations are next solved with the constant surface charge density condition on the particle and the imposed external field at the outer boundaries. Using the volumetric charge density and the electric field distribution obtained from the above solution, the Navier–Stokes and continuity equations are solved to obtain the fluid velocity field. Once the fluid velocity field is calculated, it is substituted back in the Poisson and Nernst–Planck equations to evaluate the updated electric potential and ion concentrations. The sequential solution of the Poisson, Nernst–Planck, and Navier–Stokes equations in the manner described above is continued until the solutions (velocity, electric potential, and concentrations of ions) converge to within a specified tolerance. Upon convergence of the pertinent variables to within the preset tolerance, the scaled far-field fluid axial velocity (evaluated at any location of the outer boundary), ∗ v∞ becomes equal and opposite to the scaled electrophoretic velocity, u∗p , of the parti∗ = −u∗p ) . The particle electrophoretic mobility is thus directly determined cle (i.e., v∞ from the scaled far field fluid velocity.

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TABLE 14.8. Parameter Values Employed in the Model for Calculation of Electrophoretic Mobility. Parameter

Value/Range 6.954 × 10−10 C2 /Nm2 −1 to −7

Solvent permittivity, ǫ Scaled particle surface potential, p = zeζp /kB T Particle radius Scaled particle radius, κap Ion valence, |zi | Ion diffusivity, D Temperature Fluid density, ρ Fluid viscosity, µ Applied axial electric field, E∞

10−7 m 1 to 30 1 1 × 10−9 m2 /s 298 K 1000 kg/m3 0.001 N.s/m2 104 V/m

The electrophoretic mobility, or the particle velocity per unit applied electric field, was written in Chapter 9 in context of Henry’s solution as η=

up 2 ǫζp = f (κap ) E∞ 3 µ

(14.76)

where f (κap ) is Henry’s function, given by Eq. (9.85). In terms of the scaled velocity, ∗ u∗p = (µap /ǫ)(ze/kB T )2 up , and the scaled electric field, β∞ = ap zeE∞ /(kB T ), one can express the mobility as ∗

η =

3u∗p ∗ 2β∞

3µ = 2ǫ



ze kB T



up 3µ = E∞ 2ǫ



ze kB T



η=

zeζp f (κap ) kB T

(14.77)

We will use the scaled notations of Eq. (14.77) to represent the numerical results pertaining to Henry’s function and the mobilities calculated by O’Brien and White (1978). The numerical procedures applied to solve the simplified models pertaining to the finite element implementations of Henry’s approach and the perturbation approach are briefly discussed here. In both cases, the Poisson–Boltzmann equation is first solved to determine the equilibrium electric potential and ion concentration distributions. Following this, in Henry’s approach, the Laplace equation is solved using the applied electric field as a boundary condition. The total electric potential is obtained by adding the equilibrium potential and the potential from the Laplace equation. Following this, the Navier–Stokes equation is solved to determine the velocity distribution. No iterations are required in this solution. In the perturbation analysis based formulation, the perturbed Poisson and Nernst–Planck equations are first solved using the equilibrium potential and ion concentration distributions. Following this, the perturbed Navier– Stokes equation is solved. Generally a single iteration is sufficient for the solution of these perturbed equations, since the perturbed-Nernst–Planck equations are linear. However, as we use the Navier–Stokes instead of the Stokes equation, a slight non-linearity exists that can be resolved through a couple of iterations. In subsequent

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discussions, we will refer to these numerical solutions as Henry’s numerical solution and numerical solution of perturbation equations, respectively. All three types of models were solved using the same finite element mesh, same sets of model parameters, and same numerical precision. The requirement of a highly refined mesh to capture the rapid decay of the electric potential near the particle surface, particularly for large surface potentials and κap imposes restrictions on the range of parameter values that can be explored using the numerical model described here. In particular, the procedure delineated here imposes serious demands on the number of elements for large values of κap , and accordingly, most of the simulations are confined to a small range 1 < κap < 30. 14.4.3 Representative Simulation Results We now present a comparison of the electrophoretic mobility predictions from the numerical solution with the results of Henry (1931) and O’Brien and White (1978) for a constant surface potential particle3 undergoing electrophoresis in presence of an external electric field. The purpose of these comparisons is to simultaneously validate the numerical results, and show how the analytical results of Henry (1931), although approximate, provides a remarkably good prediction of the electrophoretic mobility as long as the assumptions made in the analytical approach remain valid. Figure 14.22 shows a comparison of the function f (κap ) obtained numerically (symbols), and from Eq. (9.85) (line). The numerical simulations were conducted for a particle with a scaled surface potential of p = −1. A range of scaled particle radii (1 ≤ κap ≤ 30) was studied numerically. The numerical simulations were conducted using two approaches, namely, by direct solution of complete set of governing equations (Poisson, Nernst–Planck, and Navier–Stokes), as well as the set of simplified equations numerically emulating the methodology of Henry (1931). As seen from Figure 14.22, the numerical predictions of Henry’s function obtained from the simplified numerical implementation of Henry’s methodology (open circles) are virtually identical to the analytical values predicted using Eq. (9.85). The solid symbols, representing the results of the complete numerical simulation, on the other hand, provide a slightly lower mobility. The methodology adopted by Henry (1931) to arrive at the electrophoretic mobility was quite different from the methodology employed in the complete numerical simulations. First, Henry used the linearized Poisson– Boltzmann equation in his perturbation analysis. Secondly, he assumed an additional Laplace equation superimposed on the Poisson–Boltzmann equation to incorporate the effect of the external field. The complete numerical procedure does not use any of these assumptions, and yet, the two results are within 3% of each other. Furthermore, 3 Although we use the terminology constant surface potential, it should be recalled that we are actually keeping the particle surface charge density fixed during the electrophoretic motion in presence of the applied field. The constant surface potential simply refers to the fact that we initially determine the particle surface charge density (in absence of any external field) for each value of κap assuming a constant surface potential. The constant surface potential also refers to the fact that the particle has a constant surface potential prior to imposing the external electric field.

14.4

ANALYSIS OF ELECTROPHORETIC MOBILITY

603

Figure 14.22. Comparison of Henry’s function for electrophoretic mobility, Eq. (9.85) (line), with the corresponding predictions obtained from the finite element simulations (symbols) for a scaled particle surface potential of p = −1. Open circles represent the numerical implementation of Henry’s formulation. Solid squares represent the full numerical solution of the governing equations.

when one numerically implements the simplifications of the general governing equations that emulates Henry’s approximations, one obtains a result that is identical to his analytic expression. This comparison provides us confidence regarding the accuracy and validity of the numerical result for electrophoretic mobility. Furthermore, the comparison underscores the fact that although Henry’s analytical result is based on completely ignoring the convective relaxation effects, it is remarkably accurate, as long as the particle surface potentials are kept small. That said, however, the deviation between Henry’s result (line) and the complete numerical solution (solid symbols) in Figure 14.22 is real, and should not be solely construed as a computational error in the numerical calculations. Even for scaled particle surface potentials of p = −1, one observes a slight effect of convective relaxation as the imposed electric field becomes large. This reduces the particle electrophoretic mobility slightly. Such a deviation was observed even by Booth (1950) when the particle surface charge density was large. A discussion of this deviation is given by Saville (1977) (see Figure 3 in Saville, 1977). We now compare the predictions of the scaled electrophoretic mobility, η∗ , obtained from the numerical calculations with the corresponding predictions of O’Brien and White, (1978) for larger particle surface potentials |p | > 1. The numerical calculations of O’Brien and White (1978) pertain to solution of the perturbation equations of Section 9.3.1. Their results incorporate the convective relaxation effects, and hence, provide a more accurate prediction of the electrophoretic mobility at higher particle surface potentials compared to Henry’s function. The comparison of the finite

604

NUMERICAL SIMULATION OF ELECTROKINETIC PHENOMENA

Figure 14.23. Variation of the scaled particle electrophoretic mobility, η∗ = 3U ∗ /2β ∗ , with scaled particle surface potential at κap = 10. Line: Digitized data from O’Brien and White (1978). Solid symbols: Complete numerical solution of the governing equations. Open triangles: Numerical solution of the perturbation equations.

element results with those of O’Brien and White (1978) is shown in Figure 14.23. The comparisons are done for κap = 10. The numerical results were determined using values of p between −1 to −7. The convergence of the numerical results become increasingly difficult at higher values of p . The solid and open symbols in Figure 14.23 represent the complete numerical simulation and the numerical solution of the perturbation equations, respectively. It is evident that the numerical results obtained using the two approaches are virtually identical. These numerical results are also in good accordance with the results of O’Brien and White (1978) (line). Once again, although the numerical techniques used in the two approaches, and the governing equations solved are quite different, since the same physical problem is solved, the results should be identical. The comparison simply shows that the rigorous numerical solution of the governing equations indeed depicts the same behavior as predicted by the perturbation approach. We will restrict our presentation of results from the numerical model to the above two comparisons, since these already illustrate the applicability of direct numerical simulations to electrokinetic phenomena associated with particle electrophoresis. One can, however, obtain substantial information from these simulations regarding the detailed maps of the velocity, electric potential, and ion concentration fields. These can lead to analysis of forces acting on the particle, the total volumetric charge in the electrolyte solution surrounding the particle, the current density distribution, and similar quantities. Although we restricted the application of the general simulation to small applied fields, one can increase the applied electric field and observe the onset of several nonlinear effects on electrophoresis as well.

14.5

CONCLUDING REMARKS

605

The above comparisons lead to the conclusion that direct numerical solutions, despite their unique approximations arising from consideration of a finite computational domain, provide sufficiently accurate predictions of electrophoretic mobility. Refinement in discretization schemes, improvement in the solution methodology, and enhanced computational resources can lead to further refinement of these numerical results. However, the “cost” of these direct numerical calculations vis a vis the approximate analytical and numerical solutions should be borne in mind when considering these simulations. As a rough estimate of the computational cost, let us consider the computational time required by the three models of electrophoresis discussed in this section. The first approach, which involves the solution of the complete set of governing equations, typically requires 22 minutes on a given computer4 to converge. The numerical implementation of the perturbation equations using the same number of elements as the complete solution requires less than 3 minutes to converge. Finally, the numerical implementation of Henry’s approach requires only a few seconds to provide the solution. In this context, using a complete numerical solution is perhaps an overkill unless an extremely large electric field is applied (such that the linear perturbation approach becomes invalid). The general numerical solution may also be warranted when different particle shapes are considered, or asymmetric electrolytes with different ionic mobilities are employed. The mathematical model for electrophoretic mobility developed in this section can be modified slightly to predict the particle mobility in a cylindrical capillary. Hsu et al. (2004) numerically solved the problem when the particle translates along the axis of a charged cylindrical capillary. The presence of the charged capillary wall imposes two modifications in the model. First, the no-slip condition applies at the capillary wall, modifying the fluid velocity profile, and secondly, the charged capillary wall modifies the electrolyte concentration distribution in the computational domain, thereby modifying the electric field around the particle. In this case, the problem cannot be solved readily using a particle fixed reference frame. Instead, one starts with a guess value for the particle velocity, up , and solves the steady state equations using this velocity. From the solution, one then calculates the electrical and hydrodynamic forces on the particle. Noting that for steady state translation of the particle, these two forces will be equal and opposite, one can develop an iterative solution scheme where different guess values of the particle velocity are used until the force balance equation is satisfied. Using this methodology, Hsu et al. (2004) studied the modification of the particle electrophoretic velocity due to the presence of the charged capillary confinement. They observed that the direction of the particle motion can be reversed by using an appropriately charged cylindrical capillary confinement. 14.5

CONCLUDING REMARKS

Finite element analysis of three problems relevant to electrokinetic transport phenomena was presented in this chapter. The key message from these exercises is that 4

An Intel Pentium IV 2.8 GHz PC with 1GB RAM.

606

NUMERICAL SIMULATION OF ELECTROKINETIC PHENOMENA

the analytical approaches discussed in the previous chapters of this book often work with remarkable precision when applied judiciously to simple geometries. Numerical solution procedures, however, become indispensable when considering more complex geometries, asymmetric electrolytes with different ionic mobilities, and multi-component electrolyte solutions. In particular, when considering electrokinetic phenomena in narrow confined domains, application of numerical techniques becomes mandatory to assess the influence of the confinement on an electrokinetic phenomenon. In this context, numerical simulations become necessary in modelling microfluidic systems, where most of the phenomena like electroosmotic flow, electrophoresis, and colloidal interactions between two charged particles are modified due to the presence of the confining walls of the microfluidic channels. The numerical simulations presented in this chapter may not be regarded by some as “elegant” approaches for treating the simple problems that they were applied to. These were rigorous implementations of the exact governing equations without any attempt at simplifying these equations. However, once these detailed numerical simulation methods are developed and validated for simple systems, they can easily be applied to more complicated systems where the governing equations cannot be readily simplified.

14.6

NOMENCLATURE

a, ap b c∞ D E E∞ Ex e F Fx fx g h = I I0 (−) I1 (−) I j∗∗ i kB L Np , N n n∞ ni

particle radius, m radius of capillary, m bulk electrolyte molar concentration, mol/L ion diffusion coefficient, m2 /s electric field, Vm−1 externally imposed electric field, Vm−1 component of electric field along x direction, Vm−1 magnitude of electronic charge, C force, N component of force along axial direction, N scaled force component along axial direction gravitational acceleration, m/s2 separation distance between two colloidal particles, m unit tensor modified Bessel function of the first kind and zero order modified Bessel function of the first kind and first order total current, A ionic flux based on number concentration, m−2 s−1 Boltzmann constant, J/K length of a capillary microchannel, m scaled concentration of positive and negative ions bulk electrolyte number concentration, m−3 number concentration of i th ionic species, m−3

14.6

n nr , n z P p px q p , qc R r r¯ T = T = Te t U Ui u u∗ u, v up u∗p ∗ v∞ x x¯ z

unit surface normal components of unit normal along r and z coordinates non-dimensional pressure pressure, Pa axial pressure gradient (−∂p/∂x), Pa/m surface charge density, Cm−2 source term in a general partial differential equation radial coordinate, m scaled radial coordinate, κr or r/ap absolute temperature, K total stress tensor (hydrostatic and Maxwell stress), N/m2 Maxwell stress tensor, N/m2 time, s non-dimensional velocity vector degrees of freedom in a finite element formulation fluid velocity vector, ms−1 scaled velocity vector components of fluid velocity vector along r and x directions, respectively, m/s particle electrophoretic velocity, m/s scaled particle electrophoretic velocity scaled far-field axial fluid velocity axial coordinate, m scaled axial coordinate, κx or x/ap valence of ion

Greek Symbols β∗ ∗ β∞ Ŵ ∇ η η∗ ξi  φ ∂ ǫ κ λ µ  p

NOMENCLATURE

scaled electric field scaled applied electric field spatial gradient of dependent variable gradient operator, m−1 electrophoretic mobility, m2V−1 s−1 ∗ scaled electrophoretic mobility, 3u∗p /2β∞ basis function used in finite element technique scaled electric potential (for external field) dependent variable of a partial differential equation electric potential (externally imposed), V computational domain boundary dielectric permittivity, C2 N−1 m−2 inverse Debye length, m−1 ratio of capillary to particle radius, b/a fluid viscosity, Pa.s scaled electric potential, zeψ/kB T scaled particle surface potential, zeζp /kB T

607

608

ψ ρ ρf σp , σc τ ζp  14.7

NUMERICAL SIMULATION OF ELECTROKINETIC PHENOMENA

electric potential, V fluid material density, kgm−3 volumetric free charge density, Cm−3 scaled surface charge density dimensionless time zeta potential on a particle surface, V computational domain PROBLEMS

14.1. The Poisson–Boltzmann equation was solved numerically in Section 14.2 for a symmetric electrolyte. Generalize the model for an asymmetric electrolyte (z1 : z2 ). 14.2. Develop a numerical model for electrokinetic flow of an electrolyte through a slit microchannel. To render the model two-dimensional, assume the channel width to be much larger than the channel height. Solve the general electrochemical transport equations numerically, and compare the results with the analytical results of Chapter 8 for a slit-microchannel. 14.3. During the treatment of electrophoresis of a spherical particle using the perturbation analysis (Section 14.4.1), the perturbation equation for the ion concentration fields was obtained as Eq. (14.75). In this equation, the dependent variable is the perturbed ion concentration δNi . Note that in this equation, all the quantities with superscript “eq” are known functions. Now, suppose you have a computer program that solves the steady-state convection diffusion equation with a source/sink term, expressed as −∇ · [−D∇c + uc] = R where D is the diffusion coefficient, c is the dependent variable (concentration), u is the fluid velocity vector, and R is the source/sink term. Rewrite Eq. (14.75) to recast it in the form of the above convection diffusion equation, where c ≡ δNi . In this case, what will be the functions u and R? 14.8

REFERENCES

Anderson, J. L., and Malone, D. M., Mechanism of osmotic flow in porous membranes, Biophys J., 14, 957–982, (1974). Anderson, J. L., Colloid transport by interfacial forces, Ann. Rev. Fluid Mech., 21, 61–99, (1989). Basu, S., and Sharma, M. M., An improved space-charge model for flow through charged microporous membranes, J. Membrane Sci., 124, 77–91, (1997). Bhattacharjee, S., Chen, J. C., and Elimelech, M., Coupled model of concentration polarization and pore transport in crossflow nanofiltration, AIChE J., 47, 2733–2745, (2001).

14.8

REFERENCES

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Booth, F., The cataphoresis of spherical, solid nonconducting particles in a symmetrical electrolyte, Proc. Roy. Soc. Lond. Ser. A, 203, 514–533, (1950). Bowen, W. R., and Sharif,A. O., Long-range electrostatic attraction between like-charge spheres in a charged pore, Nature, 393, 663–665, (1998). Bowen, W. R., and Sharif, A. O., Long-range electrostatic attraction between like-charge spheres in a charged pore (vol. 393, p. 663, 1998), Nature, 402, 841–841, (1999). Burgreen, D., and Nakache, F. R., Electrokinetic flow in ultrafine capillary slits, J. Phys. Chem., 68, 1084, (1964). Carnie, S. L., Chan, D. Y. C., and Stankovich, J., Computation of forces between spherical colloidal particles – nonlinear Poisson–Boltzmann theory, J. Colloid Interface Sci., 165, 116–128, (1994). Cwirko, E. H., and Carbonell, R. G., Transport of electrolytes in charged pores – analysis using the method of spatial averaging, J. Colloid Interface Sci., 129, 513–531, (1989). Daiguji, H., Yang, P., and Majumdar, A., Ion transport in nanofluidic channels, Nano Lett., 4, 137–142, (2004). Das, P. K., Bhattacharjee, S., and Moussa, W., Electrostatic double layer force between two spherical particles in a straight cylindrical capillary: finite element analysis, Langmuir, 19, 4162–4172, (2003). Das, P. K., and Bhattacharjee, S., Electrostatic double layer interaction between spherical particles inside a rough capillary, J. Colloid Interface Sci., 273, 278–290, (2004). Das, P. K., and Bhattacharjee, S., Finite element estimation of electrostatic double layer interaction between colloidal particles inside a rough cylindrical capillary: effect of charging behavior, Colloids Surf. A, 256, 91–103, (2005). Deen, W. M., Hindered transport of large molecules in liquid-filled pores, AIChE J., 33, 1409– 1425, (1987). Glendinning, A. B., and Russel, W. B., The electrostatic repulsion between charged spheres from exact solutions to the linearized Poisson–Boltzmann equation, J. Colloid Interface Sci., 93, 95–104, (1983). Grant, M. L., and Saville, D. A., Electrostatic interactions between a nonuniformly charged sphere and a charged surface, J. Colloid Interface Sci., 171, 35–45, (1995). Gray, J. J., Chiang, B., and Bonnecaze, R. T., Colloidal particles – origin of anomalous multibody interactions, Nature, 402, 750, (1999). Gross, R. J., and Osterle, J. F., Membrane transport characteristic of ultrafine capillaries, J. Chem. Phys., 49, 228, (1968). Hall, M. S., Starov, V. M., and Lloyd, D. R., Reverse osmosis of multicomponent electrolyte solutions. 1. Theoretical development, J. Membrane Sci., 128, 23–37, (1997). Helmholtz, H. V., Studien uber elctrische grenschichten, Ann. der Physik und Chimie, 7, 337–387, (1879). Henry, D. C., The cataphoresis of suspended particles, Part 1. The equation of cataphoresis, Proc. Roy. Soc. Lond., 133A, 106–129, (1931). Hogg, R. I., Healy, T. W., and Fuerstenau, D. W., Mutual coagulation of colloidal dispersions, Trans. Faraday Soc., 62, 1638–1651, (1966). Hsu, J.-P., and Hung, S.-H., Electrophoresis of a charge-regulated spheroid along the axis of an uncharged cylindrical pore, J. Colloid Interface Sci., 264, 121–127, (2003).

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Hsu, J.-P., Ku, M.-H., and Kao, C.-Y., Electrophoresis of a spherical particle along the axis of a cylindrical pore: effect of electroosmotic flow, J. Colloid Interface Sci., 276, 248–254, (2004). Hunter, R. J., Zeta Potential in Colloid Science, Academic Press, London, (1981). Jacazio, G., Probstein, R. F., Sonin, A. A., andYung, D., Electrokinetic salt rejection in hyperfiltration through porous materials: Theory and experiments, J. Phys. Chem., 76, 4015–4023, (1972). Keh, H. J., and Chen, S. B., Particle interactions in electrophoresis. 1. Motion of 2 spheres along their line of centers, J. Colloid Interface Sci., 130, 542–555, (1989a). Keh, H. J., and Chen, S. B., Particle interactions in electrophoresis. 2. Motion of 2 spheres normal to their line of centers, J. Colloid Interface Sci., 130, 556–567, (1989b). Keh, H. J., and Tseng, H. C., Transient electrokinetic flow in fine capillaries, J. Colloid Interface Sci., 242, 450–459, (2001). Larsen A. E., and Grier, D. G., Like-charge attractions in metastable colloidal crystallites, Nature, 385, 230–233, (1997). Levine, S., Marriott, J. R., Neale, G., and Epstein, N., Theory of electrokinetic flow in fine cylindrical capillaries at high zeta-potentials, J. Colloid Interface Sci., 52, 136–149, (1975). Lewis, A. F., and Myers, R. R., The efficiency of streaming potential generation, J. Phys. Chem., 64, 1338–1339, (1960). Li, D. Q., Electrokinetics in Microfluidics, Elsevier, Amsterdam, (2004). Mansouri, A., Scheuerman, C., Bhattacharjee, S., Kwok, D. Y., and Kostiuk, L. W., Transient streaming potential in a finite length capillary, J. Colloid Interface Sci., 292, 567–580, (2005). Morrison, F. A., Transient electrophoresis of a dielectric sphere, J. Colloid Interface Sci., 29, 687, (1969). O’Brien, R. W., and White, L. R., Electrophoretic mobility of a spherical colloidal particle, J. Chem. Soc. Faraday Trans., II, 74, 1607–1626, (1978). Osterle, J. F., Unified treatment of thermodynamics of steady-state energy conversion, Appl. Sci. Res. A, 12, 425, (1964). Rice, C. L., and Whitehead, R., Electrokinetic flow in a narrow cylindrical capillary, J. Phys. Chem., 69 , 4017–4024, (1965). Sader, J. E., and Chan, D. Y. C., Electrical double-layer interaction between charged particles near surfaces and in confined geometries, J. Colloid Interface Sci., 218, 423–432, (1999). Sader, J. E., and Chan, D.Y. C., Long-range electrostatic attractions between identically charged particles in confined geometries and the Poisson–Boltzmann theory, Langmuir, 16, 324– 331, (2000). Sasidhar, V., and Ruckenstein, E., Electrolyte osmosis through capillaries, J. Colloid Interface Sci., 82, 439–457, (1981). Sasidhar, V., and Ruckenstein, E., Anomalous effects during electrolyte osmosis across charged porous membranes, J. Colloid Interface Sci., 85, 332–362, (1982). Saville, D. A., Electrokinetic effects with small particles, Ann. Rev. Fluid Mech., 9, 321–337, (1977). Shugai, A. A., Carnie, S. L., Chan, D. Y. C., and Anderson, J. L., Electrophoretic motion of two spherical particles with thick double layers, J. Colloid Interface Sci., 191, 357–371, (1997).

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Shugai, A. A., and Carnie, S. L., Electrophoretic motion of a spherical particle with a thick double layer in bounded flows, J. Colloid Interface Sci., 213, 298–315, (1999). Smit, J. A. M., Reverse-osmosis in charged membranes – analytical predictions from the spacecharge model, J. Colloid Interface Sci., 132, 413–424, (1989). Smith III, F. G., and Deen, W. M., Electrostatic double-layer interactions for spherical colloids in cylindrical pores, J. Colloid Interface Sci., 78, 444–465, (1980). Smoluchowski, M. von, Contribution a la theorie de l’endosmose electrique et de quelques phenomenes correlatifs, Bull. International de l’Academie des Sciences de Cracovie, 8, 182–200, (1903). Stankovich, J., and Carnie, S. L., Electrical double layer interaction between dissimilar spherical colloidal particles and between a sphere and a plate: Nonlinear Poisson–Boltzmann theory, Langmuir, 12, 1453–1461, (1996). Teubner, M., The motion of charged colloidal particles in electric fields, J. Chem. Phys., 76, 5564–5573, (1982). Yang, R. J., Fu, L. M., and Hwang, C. C., Electroosmotic entry flow in a microchannel, J. Colloid Interface Sci., 244, 173–179, (2001). Warszynski, P., and Adamczyk, Z., Calculations of double-layer electrostatic interactions for the sphere/plane geometry, J. Colloid Interface Sci., 187, 283–295, (1997). Wiersema, P. H., Loeb, A. L., and Overbeek, J. Th. G., Calculation of the electrophoretic mobility of a spherical colloid particle, J. Colloid Interface Sci., 22, 78–99, (1966). Ye, C., and Li, D. Q., Electrophoretic motion of two spherical particles in a rectangular microchannel, Microfluid. Nanofluid., 1, 52–61, (2004). Zienkiewicz, O. C., and Taylor, R. L., The Finite Element Method, 5th ed., McGraw-Hill, New York, (1989).

CHAPTER 15

ELECTROKINETIC APPLICATIONS

15.1

INTRODUCTION

We shall consider in this chapter some selected applications of electrokinetic transport phenomena to illustrate the influence of electric double layer and other colloidal forces on the transport of ions, charged particles, and liquids. Electrokinetic transport phenomena play an important role in many industrial processes, waste water treatment, and biological and physiological functions. Examples of selected applications will be presented spanning mechanisms of salt rejection by reverse osmosis membranes, iontophoretic drug delivery, electrokinetic management of hazardous wastes, prediction of transport properties of colloidal dispersions, application of electrokinetic and colloidal phenomena in liberation of crude petroleum from mined oil (or tar) sands, and application of electrokinetics in microfluidic systems.

15.2 ELECTROKINETIC SALT REJECTION IN POROUS MEDIA AND MEMBRANES Clays, various porous solids, and membranes have the property of partially rejecting salt when a saline solution filters through. In other words, the salt concentration in the filtrate is lower than that in the saline solution upstream of the porous medium or membrane. In order to conserve salt mass balance, the porous solid acts as a filter by not allowing the salt to pass through. Analysis of salt transport and rejection Electrokinetic and Colloid Transport Phenomena, by Jacob H. Masliyah and Subir Bhattacharjee Copyright © 2006 John Wiley & Sons, Inc.

613

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through porous media or membranes can be performed by assuming the medium to comprise of a bundle of circular cylindrical capillaries. The physical mechanism of salt rejection in a charged circular cylindrical capillary can be modeled using electrokinetic principles discussed in this book. The surface charge on the pore walls gives rise to a to a electric potential field which extends to a distance comparable to the Debye length into the capillary pore. Salt rejection analysis for a single idealized circular cylindrical pore can be extended to the actual porous medium by relating the porosity and thickness of the medium to the capillary dimensions. It should be recognized that salt rejection in a capillary is a reverse osmosis phenomenon where flow of a liquid occurs from high to low salt concentrations. Such a flow occurs under the influence of an imposed pressure gradient. The term hyperfiltration is also used for the salt rejection phenomenon to be discussed in this section. Jacazio et al. (1972) give an excellent account of the salt rejection phenomenon in a circular cylindrical capillary. Their analysis is presented here to demonstrate the application of electrokinetic principles. As in the case of electroosmosis, the total electric potential
(r, x) is assumed to be a linear combination of the induced potential φ(x) and the electric double layer potential ψ(r, x):
(r, x) = φ(x) + ψ(r, x)

(15.1)

where r and x represent the radial and axial directions, respectively. Boltzmann equilibrium is assumed, giving ¯ x)] c± (r, x) = f± (x) exp[∓ψ(r,

(15.2)

where ψ¯ = zFψ/Rg T and f± (x) is a function related to the ionic concentration. For the case where the inverse Debye length is very small compared to the capillary radius, the potential in the central core of the capillary depends only on the axial direction, x, leading to f± (x) = c(x)

(15.3)

¯ x)] c± (r, x) = c(x) exp[∓ψ(r,

(15.4)

and Eq. (15.2) becomes

The Poisson equation in cylindrical coordinates is 1 ∂ r ∂r


 ρf ∂
∂ 2
=− r + ∂r ∂x 2 ǫ

(15.5)

leading to 1 ∂ λ R ∂R 2




∂ ψ¯ R ∂R



¯ = sinh(ψ)

(15.6)

15.2

ELECTROKINETIC SALT REJECTION IN POROUS MEDIA AND MEMBRANES

615

where ∂ 2 ψ¯ ∂ 2 ψ¯ ≪ 2 ∂x ∂r 2

(15.7)

∂ 2φ =0 ∂x 2

(15.8)

r a

(15.9)

λD a

(15.10)

R= λ= and λD =




ǫRg T 2z2 F 2 c(x)

1/2

(15.11)

Equation (15.6) can be solved subject to the boundary conditions at R = 1

¯ x) = ψ¯ w ψ(r,

(capillary wall)

(15.12)

and ∂ ψ¯ =0 ∂R

at R = 0

(axis of symmetry)

(15.13)

It should be noted that the Debye length λD is a function of the axial position. The solution of Eq. (15.6) reflects the variation of λ with axial position. For a given axial position, the solution of Eq. (15.6) provides the radial distribution of the potential ¯ x). ψ(r, Jacazio et al. (1972) assumed that the fluid flow within the capillary is given by the Poiseuille relation as ux (r) = 2V (1 − R 2 )

(15.14)

where V is the average fluid velocity and ux (r) is the axial fluid velocity at radial position r. Using the fact that at any axial location, the total current is zero in steady state process, Jacazio et al. (1972) arrived at the axial ionic concentration equation


H Pe



d c¯ − c¯ = −c¯2 G d x¯

where x¯ = x/L,

dimensional axial distance

P e = V L/D∞ , c¯ = c(x)/c1 ,

flow Peclet number dimensionless axial concentration

(15.15)

616

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c1 is the feed salt concentration and L is the capillary length. The exit salt concentration c2 is given by c2 =

Js (2π a 2 V )

(15.16)

Here, Js (mol/s) is the total flux of the dissociated salt molecules. The salt rejection coefficient is defined as c2 Rr = 1 − (15.17) c1 Rr is equal to unity for complete salt rejection by the capillary tube and zero for no rejection. The coefficients H and G of Eq. (15.15) are functions of radial position, ψ¯ and λ. The initial condition for Eq. (15.15) is c¯ = 1 at x¯ = 0. The evaluation of Rr is obtained by simultaneous solution of Eq. (15.6) and (15.15). Solution details are provided by Jacazio et al. (1972). It is clear from Eq. (15.15) that the salt rejection is a function of surface charge, Peclet number and Debye length. Usually, the Debye length at the capillary inlet is specified. The salt rejection coefficient for a cylindrical pore as a function of the Peclet number for different values of the wall potential is shown in Figure 15.1. Here, λ1 represents the value of λ evaluated using the bulk molar feed concentration, c1 . Clearly, the salt rejection increases with an increasing Peclet number, and becomes constant as P e → ∞. At small, P e, diffusion is dominant with the result that there is a lower rejection. At large values of P e, the rejection increases for higher surface potentials on the pore wall.

Figure 15.1. Salt rejection coefficient of a cylindrical pore with a constant surface potential for a large Debye length as a function of the Peclet number, and comparison with experiments on compacted clays (Jacazio et al., 1972).

15.3

ELECTROOSMOTIC CONTROL OF HAZARDOUS WASTES

617

Figure 15.2. Salt rejection coefficient, Rr , of a cylindrical capillary with constant surface potential for a large Peclet number as a function of the dimensionless Debye length, λ1 , and comparison with experiments on compacted clays (Jacazio et al., 1972).

For P e ≫ 1, Figure 15.2 shows the variation of salt rejection with the inlet Debye length λ1 . The salt rejection coefficient approaches zero as the Debye length approaches zero. As salt rejection takes place entirely within the Debye sheath, a small λ1 would indicate that nearly the entire pore has ψ¯ = 0 and the electric double layer is confined to an area close to the capillary surface. Consequently, no salt rejection is expected to occur. For larger λ1 , the electric double layer extends to the entire capillary pore and rejection is enhanced. Clearly, Figure 15.2 indicates that higher salt rejection is expected for a saline feed having quite low salt concentrations. The experimental results with clays and cellophane membranes agree quite well with the capillary model put forward by Jacazio et al. as shown in Figures 15.1 and 15.2. Although a highly simplified model was used for prediction of salt rejection, it is evident from the example that electrokinetics offers an excellent means to understand the experimental results on salt rejection.

15.3

ELECTROOSMOTIC CONTROL OF HAZARDOUS WASTES

Electroosmosis entails migration of a liquid through a porous charged medium under the influence of an applied electric field. Electroosmosis has been used for the dewatering and consolidation of soils, mine tailings, and waste sludges. As fluid flow occurs due to an applied electric field, appropriate placement of electrodes would direct the fluid flow in a controlled manner. Consequently, it is possible to direct the flow of specially injected fluids or in-situ fluids in a fashion that would divert groundwater from a spill site, direct a chemical sealant toward a waste site, or dewater the region surrounding a hazardous waste area. The various modes

618

ELECTROKINETIC APPLICATIONS

Figure 15.3. Soil dewatering close to a contaminated site (Renaud and Probstein, 1987).

of electroosmotic control of hazardous wastes are given below. They are detailed by Renaud and Probstein (1987). A case of water drainage close to a hazardous waste site is shown in Figure 15.3. Here, the groundwater level is lowered under the waste site by implanting positive electrodes (anodes) around the waste area. The cathodes are located further away from the waste site. For negatively charged porous media, the salt bearing groundwater flows from the anode to the cathode, thus lowering the groundwater level in the immediate vicinity of the waste site. Figure 15.4 shows that chemical sealants injected at the anode can be directed away from the hazardous waste and thus can isolate the waste site from the groundwater flow. Renaud and Probstein (1987) discussed in detail the case where the groundwater flow toward a contaminated waste site is controlled and directed away from the contaminated area. This control is achieved by creating an adverse pressure gradient surrounding the waste site by the appropriate placements of the electrodes as shown

15.4

IONTOPHORETIC DELIVERY OF DRUGS

619

Figure 15.4. Chemical sealant controlled injection: (a) groundwater flow prior to sealant injection, and (b) groundwater flow after sealant injection (Renaud and Probstein, 1987).

Figure 15.5. Contours of constant pressure near a waste site due to the placement of electrodes (adapted from Renaud and Probstein, 1987).

in Figure 15.5. It was shown that a large adverse pressure gradient can be easily maintained through the use of the applied electrical potential.

15.4

IONTOPHORETIC DELIVERY OF DRUGS

Transdermal (skin) delivery of drugs has gained increasing importance in recent years since this route of drug administration bypasses gastrointestinal degradation and hepatic (liver) metabolism. Iontophoresis, a process which causes an increase in the migration of ionic species into the skin or tissue under a gradient of electrical potential, is used to enhance the penetration of charge molecules (Liu et al., 1988). Thus,

620

ELECTROKINETIC APPLICATIONS

iontophoresis involves the transfer of ions or charged molecules (e.g., insulin, pilocarpine) into the body by an electric field. Ions with a positive charge are driven into the skin at the anode and those with negative charges, at the cathode (Banga and Chien, 1988). Reviews discussing iontophoresis are given by Harris (1967) and Banga and Chien (1988). To a large extent, iontophoresis is similar to electrophoresis where with iontophoresis one deals with the transport of ions rather than colloidal particles. Skin manifests a large impedance to charged molecules which are transmitted through the skin under an applied electric field. The stratum corneum (outer skin layer) is the least conductive layer of the skin. Skin also has hair follicles and sweat ducts that can provide a possible pathway for the migration and diffusion of ions across the skin (Chien, 1982). Under the influence of an electric field, ionic species can penetrate the skin via the hair follicles and sweat ducts, which are referred to as “shunt” (Siddiqui, et al., 1985, 1987). Liu et al. (1988) discussed the electrical properties of the stratum corneum. They indicated that the stratum corneum has two important properties that influence iontophoresis. First, the stratum corneum can be polarized by a direct electric field (dc). Secondly, its impedance changes with the frequency of the applied field, e.g., an electric field alternating between zero and a positive value with a given waveform. Consequently, the skin can be represented by an equivalent electrical analogue as shown in Figure 15.6. The stratum corneum is represented by a combination of resistive and a capacitative component which is a function of the pulse frequency. Yamamoto and Yamamoto (1976, 1978) found that the impedance of human skin decreases with the increase in the pulse frequency. The viable skin is represented by pure resistance Rvs , which does not change with the pulse frequency. From the above characteristics of the stratum corneum, it becomes clear that when a direct electric field is utilized to enhance the penetration of charged molecules, electrochemical polarization becomes established in the skin. The induces polarization operates against the applied electrical field and reduces the current density. As pointed

Figure 15.6. Equivalent circuit of skin impedance where Rvs and Rsc are the resistors for the viable skin and stratum corneum, respectively. Csc is the capacitance for the stratum corneum (Liu et al., 1988).

15.4

IONTOPHORETIC DELIVERY OF DRUGS

621

out by Liu et al. (1988), the polarization of the stratum corneum is similar to the charging of a capacitor Csc as shown in Figure 15.6 with an initial current i. The input current decays exponentially across the stratum corneum when a constant dc voltage is applied. As the current is due to ionic movement in the skin, it is clear that the movement of the ionic species through the skin is expected to decay when a dc voltage is applied. To avoid polarization of the stratum corneum, a pulse dc voltage is normally used in the application of iontophoresis. This pulse mode is a dc voltage that periodically alternates between “on” and “off” for the applied voltage. When the voltage is “on”, the charged molecules penetrate the skin while the stratum corneum is being polarized. During the “off” period, no ionic penetration takes place and the stratum corneum becomes depolarized. The manner by which the “on” and “off” cycles are administered controls the rate of the ionic species penetration into the skin. Iontophoretic delivery of insulin to diabetic rats was investigated by Liu et al. (1988). The effectiveness of insulin delivery was monitored by measurement of the blood glucose. The lowering of the blood glucose was indicative of insulin delivery. Figure 15.7 shows the variation of the blood glucose level with time due to application of simple dc voltage and pulse dc voltage. The experiments were conducted on diabetic hairless rats. The dc voltage mode gave a slight initial change in blood glucose level (BGL); however, little change in BGL occurred after four hours. This is due to the polarization of the stratum corneum. However, pulsed dc voltage shows excellent penetration of the insulin into the rat’s blood stream. An electrokinetic model has been proposed by Schwendeman et al. (1992) to simulate iontophoresis.

Figure 15.7. Effect of delivery mode on blood glucose levels (BGL) in diabetic hairless rats treated with transdermal periodic iontophoretic system at 2 mA (0.33 mA/cm2 ) for 40 min: , simple dc; •, pulse dc (2000 Hz) (Liu et al., 1988).

622

15.5

ELECTROKINETIC APPLICATIONS

FLOTATION OF OIL DROPLETS AND FINE PARTICLES

Rising air bubbles have been used in the mineral processing industries to float valuable minerals from solid suspensions. As a bubble rises within the solids suspension, the fine solid particles are separated by adhering to the bubble surface. Such a separation technique has been shown to be quite effective in removing oil droplets from oil wastewaters (Hung, 1978; Van Ham et al., 1983; Pal and Masliyah, 1990). In order to analyze the deposition process between an air bubble and an oil droplet or a solid particle, it becomes necessary to measure the zeta potentials on the air bubbles, solid particles, and oil droplets. To this end, Okada and Akagi (1987) developed an apparatus to measure the zeta potential of air bubbles. Figure 15.8 shows a schematic diagram of the experimental setup of Okada and Akagi (1987). The apparatus is of the microelectrophoresis type and the zeta potential is determined by measuring the electrophoretic velocity of the bubbles. The measuring system consists of a microscope, an electrophoresis cell, and a video camera attached to a recorder and a monitor. The air bubbles were generated by releasing the pressure of water containing dissolved air. The bubble size was 20–40 µm. The experimental results for the air bubble zeta potential are shown in Figure 15.9 for the case of an aqueous cationic surfactant cetyl trimethylammonium bromide (CTAB) solution (5 × 10−5 mol/L) containing 0.5 vol % ethanol. Clearly, the zeta potential of the air bubbles is a strong function of the ionic strength of the solution.

Figure 15.8. Schematic diagram of experimental apparatus used to measure zeta potential of air bubbles (Okada and Akagi, 1987).

15.5

FLOTATION OF OIL DROPLETS AND FINE PARTICLES

623

Figure 15.9. Effect of concentration of Na2 SO4 electrolyte on the zeta potential of an air bubble at 25◦ C (Okada and Akagi, 1987).

This is similar for the case of the zeta potential dependence on ionic strength for liquids and solid surfaces. Okada et al. (1988) carried out further zeta potential measurements on heavy oil (ap = 1.4 × 10−6 m) and air bubbles in an anionic surfactant, sodium dodecyl sulfate (SDS) (1 × 10−4 mol/L) in the presence of different electrolytes (Figure 15.10). The

Figure 15.10. Variation of the zeta potential of air bubbles, ζc , and oil droplets, ζp , with electrolyte molarity (Okada et al., 1988).

624

ELECTROKINETIC APPLICATIONS

zeta potentials are both a function of electrolyte type and concentration. Yoon and Yordan (1986) gave extensive measurements of bubble zeta potentials and the effect of various surfactants. Okada et al. (1990a,b) carried out flotation experiments for oil droplets. They measured the flotation efficiency ηT where ηT = 1 −

cf ci

(15.18)

with ci and cf being the initial and final oil droplet concentration, respectively, in the flotation experiments. Okada et al. (1990a,b) also carried out a trajectory analysis similar to that discussed in Chapter 13. For the case of flotation of oil droplets, they found that maximum floatability was achieved when the value of the dimensionless adhesion number m = 4π ǫζp ζc /κa was in the range of m < 1.0. Their dimensionless adhesion number is related to the electrostatic repulsion number Nζ , which was defined in Eq. (13.110). Figure 15.11 shows a comparison between the calculated collection efficiency using Eq. (13.108) and the experimentally measured values. The particles employed are polystyrene latex spheres with ap = 1.47 × 10−6 m. Clearly, the zeta potentials of the particles and air bubbles play a crucial role in the collection efficiency. Fair agreement is evident between the theory and experiment. From the above, the application of the DLVO theory coupled with the hydrodynamic interaction can be effectively used to study particle flotation in a systematic manner that can be very useful in optimizing solids and oil droplets collection in an industrial process.

Figure 15.11. Comparison of calculated and experimental values of ηT , and effect of the zeta potentials of the bubble and particles on ηT in polyoxyethylene 23-lauryl ether solution (1 × 10−5 mol/L) with change of pH. The diameters of the bubbles and particles were 29.7 and 2. 95 µm, respectively (Okada et al., 1990b).

15.6

15.6

RHEOLOGY OF COLLOIDAL SUSPENSIONS

625

RHEOLOGY OF COLLOIDAL SUSPENSIONS

The rheology of a colloidal suspension is a very important property that influences the usage of the system in an industrial environment. In many situations, it is possible to alter the rheological behavior of a colloidal suspension by changing the electrolyte concentration of the continuous phase or by anchoring long-chain polymers to the surface of the colloidal particles in the suspension. 15.6.1 Historical Background When a fluid is subjected to a simple shear, the shear stress τ required to produce a shear rate (strain rate) γ˙ is given by τ = µ(γ˙ )γ˙

(15.19)

The coefficient µ(γ˙ ) is known as the apparent viscosity of the fluid. When µ(γ˙ ) is independent of the shear rate, one can write τ = µγ˙

(15.20)

where µ is a constant. Fluids obeying the formulation of Eq. (15.20) are known as Newtonian fluids. Water and simple organic liquids are Newtonian fluids. Kreiger and Eguiluz (1976) gave a tabulation of the dimensionless groups that control the rheology of a colloidal system under the influence of Brownian, viscous, and electrostatic forces. In the limiting case of negligible inertial effects, and at steady state, the relative viscosity µr of a colloidal system is given by µr =

µs = f (αp , τr , qr , κap ) µc

(15.21)

where, µs and µc are viscosities of the colloidal suspension and the continuous phase, respectively. Here αp is the colloidal particle volume fraction in the suspension. The dimensionless groups are τr , qr , and κap , where τr =

τ ap3

kB T n∞ e qr = Nq

=

γ˙ µc ap3 kB T

reduced shear stress

charge ratio

and 2z2 e2 n∞ κap = ǫkB T 

1/2

ap

dimensionless inverse Debye length

It is possible to define a particle Peclet number, P e′ as P e′ =

γ˙ ap2 D∞

(15.22)

626

ELECTROKINETIC APPLICATIONS

Making use of the Stokes–Einstein relationship for the particle diffusion coefficient, D∞ : kB T D∞ = 6π µc ap the Peclet number becomes P e′ =

6π γ˙ µc ap3 kB T

= 6π τr

(15.23)

Rescaling the particle Peclet number, P e′ , one can write P e = τr . In some literature, the particle Peclet number P e′ is used in lieu of the reduced shear τr . In the absence of electrostatic forces, the relative suspension viscosity becomes µr = f (αp , P e)

(15.24)

Here the viscosity of a suspension is affected only by the viscous forces, the Brownian motion, and the excluded volume of the particles. In such circumstances, the system is referred to as a “hard sphere” model. In the limit of infinite dilution, i.e., αp → 0, the relative viscosity of a colloidal system becomes a function of the volume fraction only. 15.6.2 Hard Sphere Model This is the case where only the viscous and Brownian motion forces are present. The relative viscosity is expected to depend only on the particle volume fraction and the particle Peclet number, P e. At a fixed volume fraction αp , it is expected that rheological data for different particle sizes and for continuous phase viscosities (µc ) collapse together when the relative suspension viscosity, µr , is plotted against P e. This is indeed the case as is shown in Figure 15.12. The plot is an S-shaped curve characteristic of uncharged particles. For given continuous medium viscosity, temperature, and particle size, the plot of Figure 15.12 indicates that the relative viscosity of a colloidal suspension is a function of the shear rate γ˙ and hence, by definition, exhibits non-Newtonian behavior. In general, due to Brownian diffusion, the relative viscosity of a suspension of colloidal particles can exhibit non-Newtonian characteristics. At low particle Peclet numbers, the Brownian motion dominates. However, at high particle Peclet numbers, the shearing force becomes dominant. In both limiting cases, the colloidal suspension exhibits Newtonian characteristics. deKruif et al. (1986) gave the limiting values for the suspension viscosities as  αp −2 µ0 = 1− µc 0.63

(15.25)

 αp −2 µ∞ = 1− µc 0.71

(15.26)

and

15.6

RHEOLOGY OF COLLOIDAL SUSPENSIONS

627

Figure 15.12. Relative viscosities versus Peclet number for a monodisperse suspension of polystyrene spheres of various sizes in different media with a fixed αp = 0.5 (Krieger, 1972).

where µ0 and µ∞ are the limiting values of the relative viscosities at P e → 0 and P e → ∞, respectively. For a given colloidal system, µ0 and µ∞ correspond to γ˙ → 0 (low shear rate) and γ˙ → ∞ (high shear rate), respectively. The S-shaped curve can be cast as (Krieger, 1972) µr = µ0 +

(µ0 − µ∞ ) 1 + 2.32P e

(15.27)

In the limit of infinite dilution, Einstein’s result for the relative viscosity is given as µr = 1 + [η]αp + O(αp2 )

(15.28)

where [η] is the intrinsic viscosity and has a value of 5/2 for the case of hard spheres. In the limit of αp → 0, there should be no dependence of the relative viscosity on the flow Peclet number. Batchelor (1977) extended Einstein’s relationship for P e ≪ 1 to give 5 µr = 1 + αp + 6.2αp2 + O(αp3 ) 2

(15.29)

Figure 15.13 shows plots for Eqs. (15.25) to (15.29). It can be observed that the Einstein and Batchelor equations are valid only for αp ≤ 0.05.

628

ELECTROKINETIC APPLICATIONS

Figure 15.13. Variation of the relative viscosity of a colloidal suspension with the particle volume fraction. Hard sphere model.

15.6.3 Electroviscous Effects Three distinct effects of electrical charge on colloidal suspension rheology have been identified (Watterson and White, 1981). The first is known as the primary electroviscous effect. It arises from the distortion of the diffuse part of the electric double layer due to shear. The second is known as the secondary electroviscous effect. It arises from interparticle interactions, which modify the particle trajectories and give rise to an increase in the effective particle excluded volume. The third effect is known as the tertiary electroviscous effect. It is due to the expansion and contraction of stabilizing polymer chains on the particle surface due to changes in the electrolyte concentrations (Hirtzel and Rajagopalan, 1985; Stein, 1985). For dilute systems, the first electroviscous effect is more significant than the second effect as it influences the coefficient of O(αp ) in Eq. (15.29). The second electroviscous effect influences the coefficient of O(αp2 ) of Eq. (15.29). Consequently, the second electroviscous effect becomes significant for moderately concentrated suspensions (Russel, 1980). The primary electroviscous effect was first identified by Smoluchowski as being due to an enhanced energy dissipation rate caused by the interaction of the diffuse part of the electric double layer with the flow field. For a dilute system, in the presence of an electric double layer, the coefficient of αp becomes [η] =

5 [1 + E] 2

(15.30)

where [1 + E] is the augmentation factor due to the presence of the electric double layer. Watterson and White (1981) gave a plot of [1 + E] as influenced by the dimensionless zeta potential (eζ /kB T ) of the particles and inverse dimensionless Debye

15.6

RHEOLOGY OF COLLOIDAL SUSPENSIONS

629

Figure 15.14. The primary electroviscous effect as a function of the electrokinetic potential in a (1 : 1) electrolyte solution (Watterson and White, 1981).

length, κap . Their plot is shown in Figure 15.14. For κap > 10, i.e., a thin electric double layer, the term (1 + E) does not exceed 1.05 for (eζ /kB T ) as high as 4 indicating little change in [η] due to the first electroviscous effect. However, when the electric double layer is thick, i.e., κap ≪ 1, the term (1 + E) becomes appreciably greater than unity and the primary electroviscous effect becomes significant. For non-dilute colloidal suspensions, the secondary electroviscous effect becomes significant where the probability of particle-particle interaction increases. Although in non-dilute colloidal systems both the primary and secondary electroviscous effects are present, the secondary effect becomes dominant at increasing particle concentrations. It is for this reason that one assumes that deviation from the hard sphere model at high concentrations is due to secondary electroviscous effects. Figure 15.15 shows variations of the relative viscosity of polystyrene latex with the continuous phase counterion concentration at different particle Peclet numbers for αp = 0.509. At high P e, say P e = 100, the relative viscosity is not sensitive to the variation of the electrolyte concentration. Here, the viscous forces are dominant. However, at lower P e values, there is a sharp increase in the relative viscosity of the colloidal suspension with decreasing electrolyte concentration where κap is smaller and the effect of the electric double layer extends further away from the particle surface. For all values of P e, there is a distinct minimum in the µr corresponding to an electrolyte concentration cmin that is independent of the Peclet number. Electrolyte concentrations greater than cmin correspond to the flocculation of the colloidal system where the repulsive force becomes weaker with the addition of electrolytes. Variation of the relative viscosity with the flow Peclet number is shown in Figure 15.16 for polystyrene latex particles, αp = 0.4, at different HCl concentrations ranging from 0 to 0.1 M. All curves tend to approach a common horizontal asymptote

630

ELECTROKINETIC APPLICATIONS

Figure 15.15. Effect of electrolyte molarity on the relative viscosity of monodisperse polystyrene latex, ap = 96 nm, αp = 0.509, pH 7.0 with 21% anionic and 79% anionic surfactants (Krieger, 1972).

with an increasing P e where the electroviscous forces are negligible in comparison to the viscous forces. According to Eq. (15.26), the asymptotic value is 5.25. At low values of P e (or low shear stress), the viscosity of the deionized latex climbs as though approaching a vertical asymptote which is indicative of a yield stress. The apparent yield stress decreases with the addition of an electrolyte. At high electrolyte

Figure 15.16. Variation of the relative viscosity of polystyrene latex (ap = 110 nm) at αp = 0.4 for various HCl concentrations (Krieger and Eguiluz, 1976).

15.6

RHEOLOGY OF COLLOIDAL SUSPENSIONS

631

Figure 15.17. Effect on the relative viscosity due to different electrolytes at the same inverse Debye length, κ (Krieger and Eguiluz, 1976).

concentrations, there is no indication of a vertical asymptote, and there is no possibility of a yield stress being present (Krieger and Eguiluz, 1976). The plot of Figure 15.16 suggests that the secondary electroviscous effects are extremely important in modifying the rheological behavior of a colloidal suspension where the relative viscosity at a given P e value can vary by several orders of magnitude. A similar plot for polystyrene latex at αp = 0.4 is shown in Figure 15.17 where additions of K2 SO4 and HCl are made. The Debye length has the same value when the concentration of HCl is double that of K2 SO4 . Figure 15.17 indicates that for given κap , the data collapse together onto one curve. The variation of the particle surface charge at the various HCl and K2 SO4 concentrations was not reported. The rheological behavior of latex particles sterically stabilized by a polymer has been investigated by Willey and Macosko (1978) and Liang et al. (1992). A plot of the relative viscosity of sterically stabilized monodisperse suspensions is given in Figure 15.18. For a given colloidal suspension, the relative viscosity increased sharply as the particle volume fraction approached its maximum packing value, a case similar to the hard sphere model. The relative viscosity data could be correlated using Dougherty and Krieger type equation (Krieger, 1972) that was derived for a hard sphere model. The relative viscosity is given by 
αp −[η]αm µr = 1 − αm

(15.31)

where αm is the maximum packing volume fraction of the particles. It should be recognized that due to the attached polymeric chains at the colloidal particle surface, the volume fraction αp is simply a nominal value. The effective volume fraction of

632

ELECTROKINETIC APPLICATIONS

Figure 15.18. Relative viscosity versus volume fraction of polystyrene latex dispersions for various particle sizes (Liang et al., 1992).

the dispersed phase is given by 
3 δ αeff = αp 1 + ap

(15.32)

where δ is the polymeric chain length, and is a function of the volume fraction αp and shear rate γ˙ . The relative viscosity data at αp = 0.3 of Willey and Macosko (1978) are plotted against P e in Figure 15.19 for a different PVC plastisol particle size and solvent quality. The collapse of the data onto a single curve would suggest that the PVC plastisols behave as an ideal colloidal suspension of rigid spheres subject to Brownian diffusion and they are little affected by the repulsive forces. The limiting relative viscosity value at P e → 0 is about 300. Comparison with the prediction from Eq. (15.25) indicates that a much greater effective volume fraction is operative than the nominal value of αp = 0.2 (Russel, 1980). The rheology of a colloidal system is clearly a strong function of the prevailing colloidal forces. By altering the relative magnitude of these forces, it is possible to obtain the desired rheological characteristic. It is for this reason, among others, that colloidal suspensions have found many applications in paints, dyestuffs, and pharmaceutical and pesticidal formulations.

15.7

BITUMEN EXTRACTION FROM OIL SANDS

We will explore in this section the application of electrokinetic phenomena to understand and appreciate a major industrial energy process provider. The theoretical

15.7

BITUMEN EXTRACTION FROM OIL SANDS

633

Figure 15.19. Relative viscosity of sterically stabilized PVC plastisols at αp = 0.2 for different particle sizes and solvents (Willey and Macosko, 1978).

analysis presented in earlier chapters will be used to shed some understanding on bitumen extraction from oil sands. Oil sands are also known as tar sands and bituminous sands. They are unconsolidated sand deposits that are impregnated with high molar mass viscous petroleum, normally referred to as bitumen. Bitumen within the oil sands ore can be thought of as being a very viscous oil embedded within a sand matrix. At room temperature, bitumen is like cold molasses having very high viscosity and it is difficult to make it flow under gravity. Bitumen must be treated in upgraders to convert it into “synthetic crude oil” before it can be fractionated by refineries to produce gasoline, heating oils, and diesel fuels. Oil sands are found throughout the world, usually in the same geographical location as conventional petroleum. The world’s two largest sources of bitumen are in Canada and in Venezuela. Canada’s bitumen resources are located almost entirely within the province of Alberta. Alberta’s oil sand deposits are grouped on the basis of geology, geography and bitumen content, (NEB-EMA, 2004) and they are found in three locations: The Athabasca, Peace River and Cold Lake regions. The bitumen deposits in these three areas are found in sedimentary formations of sand and carbonate that collectively cover six million hectares. The largest deposit in the world is in the Athabasca area in the northeast part of the province of Alberta, Canada (Camp, 1976; Oil & Gas, 2004). Based on current data, the Alberta Energy and Utilities Board (AEUB) estimates that the ultimate bitumen volume, a value that represents bitumen volume expected to be found by the time all exploration has ceased to be 400 × 109 m3 (2.5 × 1012 barrels). Of the ultimate bitumen volume-in-place, 50 × 109 m3 (315 × 109 barrels) are estimated to be recoverable using current and anticipated new technologies.

634

ELECTROKINETIC APPLICATIONS

With current technologies, it is estimated that about 5.6 × 109 m3 , which are at depths of less than 75m, are amenable to surface mining. At higher buried depths, about 22.7 × 109 m3 of the oil sands are amenable to underground type mining and in-situ (in-place) production (NEB-EMA, 2004; Oil & Gas, 2004). In 2004, the total Canadian daily bitumen production from open pit operations was about 625,000 barrels/day. The oil sands deposits are composed of mainly quartz sand, clays, connate water and associated salts and bitumen. The clays are predominately kaolinite. The bitumen content varies from 7 to 15% by weight. The mineral solids are about 80–85% by weight and the connate water is 3–5%, by weight. The mineral solids particle size varies from less than 0.1 to 300 microns. The density of bitumen is about 1050 kg/m3 with a viscosity of about 1000 Pas at room temperature. As stated by Czarnecki et al. (2005), there is a general belief, as yet to be proven, that because for the most part, the mineral sand grains in Athabasca oil sands are hydrophilic, they are also “water-wet”. In other words, the bitumen is not in direct contact with the sand grains, but instead separated from the sand grains by a nanothick water film. The presence of this water film is assumed to be the major difference between Alberta oil sands and Utah tar sands that are considered to be “oil wet”. Subsequently, bitumen from “water-wet” oil sands can be recovered using a water based extraction process where bitumen can be easily liberated from a hydrophilic or a “water-wet” surface. On the other hand, bitumen from hydrophobic or “oil-wet” sands would not be easily recovered in a water-based extraction process as it would be more difficult to dislodge bitumen that is directly attached to the sand grains. Due to the nature of the mineral solids, recovery of bitumen is achieved my mixing warm water with mined oil sand ore to liberate the bitumen with its subsequent flotation in gravity settlers. The processing temperature can vary from 35◦ C to 75◦ C. Conceptually, bitumen recovery from oil sands using water-based extraction processes as applied to open pit mining, involves the following steps (Masliyah et al., 2004): (i) Bitumen Liberation from Sand Grains: To recover bitumen from the oil sand ore, the bitumen needs first to be liberated from the sand grains. This step is controlled by the process temperature, mechanical agitation and interfacial properties. The dislodging of the bitumen from the sand grains is the first step in the bitumen recovery process. (ii) Bitumen Aeration: As bitumen density is very close to that of water, in order to recover the liberated bitumen in gravity settlers, it is necessary to aerate the liberated bitumen by having air attach to it. It is well known in the oil sands industry that oil sand ores containing high mineral fines (fines are defined as mineral solids less than 44 microns in size) processed in a slurry containing high concentrations of divalent ions give a low bitumen recovery (Masliyah et al., 2004). The divalent ions are calcium and magnesium. Conversely, oil sand ores containing low mineral fines processed in a slurry containing low concentrations of divalent ions, if not aged, would lead to a high bitumen recovery. The

15.7

635

BITUMEN EXTRACTION FROM OIL SANDS

Figure 15.20. Schematic process chart for bitumen extraction from oil sands.

hypothesis is that the presence of high mineral fines coupled with high concentration of divalent ions leads to the coating of the bitumen droplets by the mineral fines, i.e., hetero-coagulation. As these mineral fines are in general hydrophilic, finescoated bitumen droplets become less hydrophobic and would not efficiently attach to hydrophobic air bubbles. As bitumen has similar density as water, the bitumen-air attachment process is vital to reduce the bitumen density for its recovery. A poor bitumen-air attachment would lead to a low bitumen recovery. A schematic bitumen recovery from oil sands is shown in Figure 15.20. From the afore-mentioned bitumen recovery steps, it becomes essential to provide the environment for efficient bitumen liberation from the sand grains and bitumen-air attachment. Any factor that adversely affects these two steps would give a lower bitumen recovery. In this Section, zeta potential distribution measurements and colloidal force measurements will be used to show that (i) repulsion and attraction forces between bitumen and silica surfaces can lead to poor bitumen liberation from a silica surface, and (ii) undesirable mineral fines when coupled with high divalent ion concentration would lead to surface coating of solid mineral fines on a bitumen surface. 15.7.1 Zeta Potential of Oil Sand Components The origin of the surface charge depends on the surface chemistry and the electrolyte solution. For the case of bitumen-water systems, it is assumed that the natural surfactants present in the bitumen are responsible for the origin of the bitumen-water interface charge. The negative charge that is usually present at the bitumen-water interface can be explained by the dissociation of the carboxyl groups belonging to the surfactants that are naturally present in bitumen (Takamura and Chow, 1985). RCOOH + OH− ⇋ RCOO− + H2 O −

RCOOH ⇋ RCOO + H

+

(15.33) (15.34)

The presence of NaOH would modify the dissociation state and can subsequently alter the surface charge. One would expect that, with the addition of NaOH, the above reaction, Eq. (15.33), proceeds more to the right hand side, whereby the bitumen

636

ELECTROKINETIC APPLICATIONS

20

Bitumen zeta potential,mV

0 –20

KCL,mM

–40

1

–60 10 –80

100

–100 2

4

6

8

10

12

pH

Figure 15.21. Variation of bitumen zeta potential at room temperature with bulk pH at different KCl concentrations.

surface becomes more negatively charged. Industrial bitumen extraction processes utilize water with a pH of about 8.5. Figure 15.21 shows the variation of zeta potential of bitumen at room temperature as a function of potassium chloride concentration. Two observations can be made. For a given KCl molarity, the zeta potential becomes more negative at high pH and that the zeta potential of bitumen, for a given, pH, is less negative at higher KCl molarity. When a multivalent cation is present in the bulk solution (e.g., Ca2+ ), ion-binding between the carboxyl group present in bitumen and the cations takes place (Takamura and Chow, 1985; Takamura et al., 1986). RCOO− + Ca2+ ⇋ RCOOCa+

(15.35)

Here, one would expect an increase in the surface potential (less negative) with the addition of, say, CaCl2 . Figure 15.22 shows the bitumen zeta potential variation with pH at various calcium chloride concentrations. The surface charge of silica mineral-water interface can also be explained using the surface groups ionization concept of Healy and White (1978). The silica surface becomes either positively or negatively charged depending on the equilibrium associated with the following surface ionization reactions (Takamura and Isaacs, 1989) At acidic conditions: −SiOH + H+ ⇋ −SiOH+ 2 −

−

At alkaline conditions: −SiOH + OH ⇋ −SiO + H2 O

(15.36) (15.37)

Figure 15.23 shows the variations of silica zeta potential with pH for different CaCl2 concentrations.

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20

Zeta Potential,mV

0 Calcium,mM

–20

1 –40 0.1 –60 0

–80 –100 2

4

6

8

10

12

pH

Figure 15.22. Variation of bitumen zeta potential at room temperature with bulk pH for different CaCl2 concentrations. The background KCl concentration is 1 mM (Liu et al., 2003).

In the case of clays, the crystal charge is due to substitution of aluminum for silicon in the tetrahedral layer with a consequent imbalance of negative charge. For example, Al3+ may replace Si4+ at the surface of the clay, producing a negative surface charge. In this case, a point of zero surface charge, PZC, can be reached by reducing the pH. Here, the added H+ ions combine with the negative charges on the surface to form OH groups. Such a charge origin is referred to as isomorphous substitution. 20

Calcium,mM

Zeta Potential,mV

0

1 –20 0.1 –40 0

–60

–80 2

4

6

8

10

12

pH

Figure 15.23. Variation of silica zeta potential with pH: Effect of added calcium ion on silica zeta potential at room temperature. KCl at 1 mM is the background electrolyte concentration (Liu et al., 2003).

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Figure 15.24. A kaolinite clay crystal showing surface charges at a pH of, say 5.

In some clays, e.g., kaolinite, when a platelet is broken, the exposed edges contain AlOH groups which take up H+ ions to give a positively charged edge at a low pH. The edge positive charge decreases to zero as the pH is raised to about 7. The positive edge surface charge may coexist with the negatively charged basal surfaces leading to special properties. In this case, there will be no single point of zero charge, PZC, but each surface (edge and basal) has its own value. Figure 15.24 shows a typical kaolinite platelet with a negative charge at the basal surface and positive charge at the edge. The positive charge is eliminated at a pH above 7 (Hunter, 2001; Everett, 1988). The variation of zeta potential of kaolinite clays with pH at different calcium chloride molarity is shown in Figure 15.25. At low pH values, the zeta potential is positive and it is increasingly negative at higher pH values. For a given pH, it is clear that the presence of the divalent calcium ions has a very large effect on the clay zeta potential. The zeta potential becomes closer to zero at higher calcium chloride concentrations. From the zeta potential plots for the bitumen and mineral solids, it becomes clear that bitumen processability would be affected by the process water pH and electrolyte type and concentration.

10

Zeta Potential,mV

0 Calcium,mM 1

–10

0.25 –20 0

–30

–40

2

4

6

8

10

12

pH

Figure 15.25. Variation of the zeta potential of kaolinite clays with bulk pH for different calcium ion concentrations, (Liu et al., 2004a).

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15.7.2 Zeta Potential Distribution Measurements Technique From the zeta potential measurements for both bitumen and silica, it comes clear that presence of high calcium concentrations would decrease the repulsion force between silica and bitumen surface with the possibility of a net attractive force between them. Electrokinetic properties of bitumen and silica have been investigated extensively by many researchers, using conventional electrophoretic zeta meter. In the early measurements, only the average electrophoretic mobility or zeta potential for a single component system was reported. In the case of a binary mixture system, the measured average zeta potentials of the two components often either over or under-estimate the electrokinetic behavior of the system, thereby giving rise to misguided information regarding to the interpretation of interactions between the two components. With the capability of measuring electrophoretic mobility or zeta potential distributions, it is possible to identify the attraction of one component on the other in a binary suspension system. The concept for this application is described as follows. When zeta potential distributions are measured separately using emulsified bitumen droplets and silica particles, each has its own unique zeta potential distribution centered at ζB and ζS , respectively. For illustrative purposes, these two distributions are overlaid schematically in Figure 15.26(a). When bitumen and silica are mixed together under the same physicochemical conditions, the measured zeta potential

Figure 15.26. Schematic zeta potential distributions for a binary particulate component system made up of bitumen and silica: (a) zeta potential distribution of bitumen and silica measured individually; (b) bitumen and silica mixture showing no attraction; (c) partial covering of bitumen by silica and/or partial covering of silica by bitumen; and (d) strong attraction between bitumen and silica where the bitumen droplets are fully covered by the silica particles (Adapted from Liu et al., 2002).

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distribution of the resultant mixture can be interpreted in terms of the interactions between the two components. If the bitumen and silica do not interact with each other, a bimodal zeta potential distribution with two peaks centered at ζB and ζS as shown in Figure 15.26(b) is anticipated. Due to hydrodynamic interaction of moving particles at different electrophoretic mobilities, a slight shift of ζB and ζS towards each other may be observed. In the case where there is a weak attraction between the bitumen and silica particles, the measured zeta potential distribution would show several distribution peaks whose positions depend on the bitumen to silica ratio, Figure 15.26(c). When the silica particles completely cover the bitumen droplets, one peak would be observed whose location would be similar that of individual silica particles, Figure 15.26(d). The use of zeta potential distribution measurements will be illustrated later in this Section. 15.7.3 Atomic Force Microscope Technique The classical DLVO (Derjaguin–Landau–Verwey–Overbeek) theory, which considers the sum of van der Waals and electric double layer interactions, is often used to predict colloidal particle interactions. Calculation of the electrostatic double layer and van der Waals interactions is relatively straightforward for well defined systems with known system parameters such as surface potential of the interacting colloidal particles and Hamaker constant. However, in complex systems, the classical DLVO theory might not accurately predict the colloidal interactions and an extended DLVO theory has to be used, which would include repulsive hydration force for hydrophilic surfaces, attractive hydrophobic force for hydrophobic surfaces, repulsive steric force and attractive bridge force for polymer bearing surfaces. The theory for describing these additional non-DLVO forces is less well developed. Subsequently, these forces are often inferred to from the deviation of the experimentally measured colloidal forces from those predicted by the classical DLVO theory. As the DLVO theory cannot be depended upon for a complex system, direct force measurement is needed. To that end, direct force measurement with atomic force microscope, AFM is a most direct method for the measurement of the interaction forces between bitumen, silica and clays. The atomic force microscope, AFM, consists of a piezoelectric translation stage, a cantilever tip, a laser beam system, a split photodiode and a fluid cell. The working principle of AFM for colloidal force measurement can be found in the literature (Ducker et al., 1992). In bitumen extraction investigations, the probe particles are model spherical silica, and pseudo-spherical fine solids of about 5 ∼ 10 µm in diameter. A probe particle was glued with a two-component epoxy onto the tip of a short, wide beam AFM cantilever under an optical microscope. Bitumen substrates were prepared by spin-coating bitumen on silica wafers (Liu et al., 2003; Liu, 2004; Liu et al., 2004a,b). The spin-coated bitumen substrate was glued onto a magnetic plate that was mounted on the piezo-electric translation stage as shown in Figure 15.27. The cantilever substrate with a probe particle was mounted in a fluid cell. When the piezostage brings the bitumen substrate to approach or retreat from the probe particle in

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Figure 15.27. Schematic view of an atomic force microscope (AFM) showing the particle probe, the sample surface and the measuring photo diode.

the vertical direction, the force between the two surfaces causes the cantilever spring to deflect upward or downward, depending on the nature of the force between them. The deflection of the cantilever spring is detected by the position-sensitive laser beam that is focused on the upper surface of the spring cantilever, and reflected to the split photodiode through a mirror. From the displacement of the piezo-stage and the deflection of the spring cantilever, the long-range force and adhesive force (pull-off force) between the substrate and the probe particle can be obtained. Figures 15.27 to 15.29 show some illustrations of the AFM configuration. Adhesive force data can be collected under given loading force. A typical set of raw data of probing surface forces is shown in Figure 15.29. The repulsive force barrier and adhesive force are of great concern since they control the coagulation behavior between two substrates. 15.7.4 Electrokinetic Phenomena in Bitumen Recovery from Oil Sands 15.7.4.1 Bitumen-Silica Interaction: Bitumen Liberation from Sand Grains Zeta Potential Measurements: To demonstrate the interaction between colloidal bitumen droplets and silica particles, the zeta potential distributions of bitumen emulsion and silica suspension, individually and in a mixture, were measured

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Figure 15.28. AFM schematic showing (a) a model silica probe particle; (b) clay particle; and (c) a cantilever.

and the results are interpreted in terms of colloidal interactions. For illustrative purposes, only the results with and without calcium addition at solution pH of 10.5 are presented here. Let us first discuss the case of absence of calcium. As shown in the overlaid histogram of Figure 15.30(a), the zeta potential distributions measured with bitumen droplets or silica particles alone in the electrolyte solution without calcium addition, are centered at −82 and −67 mV, respectively. A similar zeta potential distribution histogram is obtained for the mixture of the two species as shown in Figure 15.30(b). The presence of two distinct distribution peaks at −82 and −69 mV indicates that bitumen droplets and silica particles in the mixture are non-coagulative, i.e., they are present separately as individual particles. Now let us include calcium in the water where 1 mM calcium chloride, in the form of calcium chloride, is added. As shown in Figure 15.31(a) the zeta potential distribution histograms for bitumen droplets and silica particles measured separately were found to be much less negative. When zeta potential distribution was measured with the mixture of the emulsified bitumen and silica particles, only a single broad distribution peak located in between the two original peaks is observed [Figure 15.31(b)]. The disappearance of the original zeta potential distribution peaks of silica and bitumen and the appearance of a new broad distribution peak in

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Figure 15.29. AFM force measurement modes: Curves a to b up to c, are in repulsion mode. Contact of a flat surface with a particle probe at point c. Curves c and d are in adhesion measuring mode. At point d, separation between the surface and probe takes place. At this location, the adhesion force is measured.

between suggest that the silica particles and bitumen droplets are hetero-coagulated to form composite aggregates. The zeta potential distribution measurements of Figures 15.30 and 15.31 clearly show that the presence of calcium ions has the impact of adhering bitumen and

Frequency (%)

(a) 20 15 10

Bitumen

Silica

5 0

Frequency (%)

(b) 20 15 10 5 0 –90

–80

–70 –60 Zeta potential,mV

–50

–40

Figure 15.30. Zeta potential distribution of bitumen emulsion and silica suspension in 1 mM KCl solution at pH 10.5 without calcium, measured (a) separately and (b) as a mixture (Liu et al., 2003).

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Frequency (%)

(a) 20 15 Silica

Bitumen

10 5

Frequency (%)

0 (b) 20 15 10 5 0 –50

–40

–30

–20

–10

0

10

Zeta potential,mV

Figure 15.31. Zeta potential distribution of bitumen emulsion and silica suspension in 1 mM KCl solution at pH 10.5 with 1 mM calcium ion addition, measured (a) separately and (b) as a mixture (Liu et al., 2003).

silica together, where hetero-coagulation takes place, a situation that would hinder the liberation of bitumen from the silica sand grains. Atomic force microscope measurements: To take into account all the prevailing colloidal forces between a bitumen surface and a silica probe, atomic force microscope measurements are conducted and they are shown in Figure 15.32 as a function of pH in the absence of calcium and in Figure 15.33 in the presence of calcium at a pH of 10.5. The solution pH is a critical operating parameter in bitumen recovery, and in most cases, the controlling parameter for surface charge. In a 1mM KCl solution, the effect of pH on the interaction forces between the bitumen surface and silica particle is shown in Figure 15.32. The measured long-range force profiles are monotonically repulsive. The repulsion force increases with increasing pH. At pH 3.5, a very weak long-range repulsive force is observed. As shown by the solid lines of Figure 15.32, at separations greater than about 2–3 nm, all the measured force profiles can be reasonably well fitted with the classical DLVO theory. The good fit shown in Figure 15.32 by the solid lines suggests that the long-range repulsive forces are predominantly from the electrostatic double layer interactions. The fitted Debye decay length ( κ −1 ) in the range of 8.5−10 nm agrees well with that calculated for a 1mM KCl solution used in the experiment, further confirming electrostatic nature of the long-range repulsive force. It should be noted that the observed repulsive force at separation distances less than 2–3 nm is inconsistent with the attractive force regime as predicted by the DLVO theory, suggesting the presence of an additional repulsive force. Although the exact reason for this contradiction is not clear, considering a 2–3 nm range, this additional repulsive force would appear to originate from brush-like surfaces or small protrusions at the bitumen/water interface, resulting in a steric type of repulsion.

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Figure 15.32. Interaction forces between bitumen and silica as a function of separation distance in 1 mM KCl solution at different solution pH. Solid lines represent the DLVO fitting using ABW S = 3.3 × 10−21 J with best-fitted decay length and Stern potential of: down triangles, pH 3.5: κ −1 = 9.4 nm, ψB = −20 mV, ψS = −25 mV; circles, pH 5.7: κ −1 = 9.4 nm, ψB = −58 mV, ψS = −48 mV; up triangles, pH 8.2: κ −1 = 9.1 nm, ψB = −76 mV, ψS = −59 mV; squares, pH 10.5: κ −1 = 8.6 nm, ψB = −83 mV, ψS = −64 mV. Force normalization is made by dividing the force by the probe particle radius (Liu et al., 2003).

A dramatic effect of calcium addition on both long-range colloidal force was observed at pH 10.5, as shown in Figure 15.33. The long-range colloidal forces changed progressively from repulsive to attractive with increasing calcium ion addition to 1 mM. Again, the long-range forces can be well fitted by the DLVO theory, indicating that the change of interaction force profiles can be simply attributed to the diminished electrical double layer forces by the specific adsorption of calcium ions. To illustrate the electrokinetic findings as illustrated by the zeta potential distributions of bitumen and silica and AFM direct measurements, the recession of an initially bitumen disc patch on a silica glass slide immersed in water is shown in Figure 15.34. The recession rate is fast at low calcium concentrations and fairly slow at higher concentrations. Moreover, the contact angle is fairly small at low calcium concentrations and it is more favorable for bitumen detachment from a silica surface. 15.7.4.2 Bitumen-Fine Hetero-Coagulation

Solids

Interaction:

Bitumen-Fine

Solids

Zeta Potential Measurements Fine Solids from Good Processing Ore: After performing bitumen extraction using a good processing ore, the mineral fine solids are collected and used for

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Figure 15.33. Interaction of normalized forces between bitumen and silica as a function of separation distance in solution containing 1 mM KCl and different concentrations of calcium ions at pH 10.5. Solid lines represent the DLVO fitting using ABW S = 3.3 × 10−21 J with bestfitted decay length and Stern potential of: squares, 0 mM CaCl2 : κ −1 = 9.1 nm, ψB = −83 mV, ψS = −64 mV; circles , 0.1 mM CaCl2 : κ −1 = 7.8 nm, ψB = −45 mV, ψS = −35 mV; up triangles, 1 mM CaCl2 : κ −1 = 4.5 nm, ψB = −38 mV, ψS = −8 mV. Force normalization is made by dividing the force by the probe particle radius (Liu et al., 2003).

Figure 15.34. Dynamic contact angle of bitumen on a glass slide at pH = 9 and 22◦ C: Effect of calcium addition (Masliyah et al., 2004).

zeta potential distribution measurements together with emulsified bitumen. The water used in these tests was that obtained from the extraction process itself. The results in Figure 15.35a show two distinct distribution peaks at −71 and −45 mV, corresponding to bitumen and fine solids, respectively. For the mixture of

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30

20

10

Bitumen

Fine Solids

(b)

30

Frequency (%)

0

20

10

0 –100

–80

–40 –60 Zeta pontential,mV

–20

0

Figure 15.35. Good processing ore: Zeta potential distributions in its corresponding extraction process water of (a) individual emulsified bitumen and fine solids suspension and (b) their mixture (Liu et al., 2005).

the bitumen and fine solids, a bimodal distribution histogram as shown in Figure 15.35b is observed. The peak values of the zeta potential distribution histogram correspond to those for the bitumen and fine solids, respectively, thereby illustrating a negligible attraction between the two components. The results confirm the negligible attraction between the fine solids and bitumen droplets as observed in industrial practice while dealing with a good processing ore. Fine Solids from Poor Processing Ore: The zeta potential distributions of bitumen droplets and mineral fines from poor processing ores are also measured separately in the corresponding process water from the extraction test itself. The results of Figure 15.36(a) show that the distribution for individual emulsified bitumen and mineral fine solids is presented by two peaks at −50 and −28 mV. Compared with Figure 15.35(a) for the case of a good processing ore, a reduction of zeta potential values for both bitumen and fine solids is observed as anticipated from the compression of electric double layer by the presence of calcium and magnesium ions in the extraction water from a poor processing ore. The zeta potential distribution of

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Frequency (%)

(a) 30

20 Bitumen

Fine Solids

10

30

Frequency (%)

0 (b)

20

10

0 –100

–80

–60 –40 –20 Zeta pontential,mV

0

Figure 15.36. Poor processing ore: Zeta potential distributions in corresponding extraction process water of (a) individual emulsified bitumen and fine solids suspension and (b) their mixture (Liu et al., 2005).

Figure 15.36(b) for the mixture of the bitumen and fine solids shows a distribution peak at a zeta potential value corresponding to that for the fine solids, with a small tail spreading towards the bitumen distribution peak. Comparing the results for the fine solids from the good processing ore as shown in Figure 15.35(b), the results here suggest a stronger attachment between the bitumen and mineral fine solids derived from a poor processing ore in its corresponding extraction process water. Atomic Force Microscope Measurements The surface force profiles measured with bitumen-fine solids pairs in their corresponding extraction process water are shown in Figure 15.37. Once again the mineral fine solids are obtained from the individual oil sand ore. The interaction force between bitumen and fine solids from a good processing ore is strongly repulsive. However, the interaction force between the bitumen and fine solids from a poor processing ore is attractive at a separation less than 10nm. The adhesion force between bitumen and fine solids pairs in their corresponding extraction process water is shown in Figure 15.38. A stronger adhesion force

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Figure 15.37. Normalized interaction forces between bitumen and fine solids collected from the oil sand ores as a function of separation distance in their corresponding extraction process water. Force normalization is made by dividing the force by the probe particle radius (Liu et al., 2004b, 2005). Solid circles: Bitumen-fine solids from good processing ore. Open circles: Bitumen-fine solids from poor processing ore. Solid line represents the classical DLVO fitting using ABW F = 6.5 × 10−21 J with best-fitted decay length and Stern potential being: κ −1 = 4.6 nm, ψB = −60 mV, ψF = −30 mV. Dotted line represents extended DLVO fitting using ABW F = 6.5 × 10−21 J with best-fitted decay length, Stern potential, κ −1 = 4.3 nm, ψB = −42 mV, ψF = −28 mV with added hydrophobic force. 40 Bitumen-Fine Solids Good Processing Ore Poor Processing Ore

Frequency (%)

30 (a) 20

(b)

10

0 0

16 12 8 4 Normalized adhesion force, mN/m

20

Figure 15.38. Distribution of normalized adhesion forces between bitumen and fine solids in their corresponding extraction process water. (a) Bitumen-fine solids collected from good processing ore and (b) Bitumen-fine solids collected from a poor processing ore. Force normalization is made by dividing the force by the probe particle radius (Liu et al., 2004b, 2005).

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between bitumen and fine solids is observed from the poor processing ore than between bitumen and fine solids from a good processing ore. The measured strong repulsive force and weak adhesion force between the bitumen and mineral fine solids, Figures 15.37 and 15.38 together with the zeta potential distributions, Figure 15.35, for the good processing ore account for the negligible slime coating of the fine solids on bitumen during bitumen extraction. The findings, from electrokinetic considerations, suggest a high bitumen flotation rate from good processing ores. A situation that is often the case in practice. The measured non-contact attractive force and strong adhesion force, Figures 15.37 and 15.38, together with the zeta potential distributions of Figure 15.36, explain the observed strong attachment between the fine solids and bitumen. The results imply a stronger coagulation of fine solids from a poor processing ore with bitumen during bitumen extraction. Such a hetero-coagulation leads to harmful slime coating. Therefore, a low bitumen recovery and poor bitumen product quality are anticipated for poor processing ores. Indeed, this is a situation experienced in an industrial oil sands operation. The electrokinetic experimental results from both the zeta potential distribution and atomic force microscope measurements shed good understanding on the industrial bitumen recovery process.

15.8

MICROFLUIDIC AND NANOFLUIDIC APPLICATIONS

Conventional complex chemical analysis, generally conducted in laboratories using large equipment, require large volumes of analytes, considerable human resources, and are time consuming. During the past decade, tremendous progress has been made toward development of technologies that pertain to miniaturization of such complex analytical procedures into extremely small microchips. Such chips are commonly referred to as microfluidic chips, lab-on-chips, or micro total analysis systems (µTAS). One can consider such chips as a subset of micro-electromechanical systems (MEMS) (Gad-el-Hak, 2002). Microfluidic chips are comprised of microchannels ranging in dimension from a few microns to less than a millimeter. The handling of fluids in such narrow channels falls under the purview of the subject of microfluidics. Microfluidic chips can be fabricated on a variety of materials, including glass, polymers, silicon, etc., employing surface micromachining procedures such as photolithography, glass etching using hydrofluoric acid (HF), molding, and embossing. Using a synergistic combination of pressure, electrical fields, interfacial and material properties of different elements in these chips, as well as, utilizing the electromechanical response of different materials to electromagnetic impulses (such as piezoelectric and dielectric polarization effects), one can perform an astounding array of fluid manipulation in these devices. The devices combine a wide range of functions including sample loading, pumping, screening, separation, mixing, and diagnostics (measurement of properties). The small volumes (typically nanoliters to microliters) of fluid

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required for the analysis, portability, and the rapid nature of the assays make microfluidics based chemical analysis a highly attractive alternative to conventional chemical analysis. With the remarkable global interest in nanotechnology during the past decade, there has been a significant resurgence of interest in nanofluidics or fluid flow in and around nanoscale objects.1 It is, however, important to note that nanofluidics existed, albeit implicitly, within the scope of many disciplines over the last century. Literature in diverse disciplines, such as biological sciences, catalysis, colloid transport and electrochemistry, membrane science, physics of fluids in confined media, and soil science, to name a few, provide a significant knowledge base in this area. The modern thrust in nanofluidics research has sometimes overlooked this enormous knowledge base, and reinvented ideas that have been existent for over a century. Common examples cited in this context (Eijkel and van den Berg, 2005) are the rediscovery of Donnan exclusion (Pu et al., 2004) and ion rejection through narrow capillaries (Daiguji et al., 2004) after nearly a century of their original incorporation in our knowledge base. Nevertheless, the promise of nanofluidics lies in the recent ability to fabricate nanoscale structures and channels with unprecedented geometrical precision, which can potentially lead to novel techniques for separation, single molecule detection, and other biological and chemical assays. In context of the continuum theory described in this book, the fluid flow behavior in microfluidic and nanofluidic systems can be addressed in light of the same physical models. The demarcations imposed between microfluidics and nanofluidics often stem from the somewhat misleading notion that the flow mechanisms are very different in these domains. However, from the point of view of classical electrohydrodynamic theories, the fluid mechanical and electrokinetic transport behavior observed over the length scales spanning nano- and microfluidics is tractable using the same theoretical principles. The only length scale at which such continuum mechanics based theories can break down is typically below a couple of nanometers, when the solvent cannot be treated as a continuum, and quantum effects might become important. Above this length scale, however, it is generally believed that continuum approaches are adequate in resolving the flow behavior. The history of modern microfluidics dates back to the work of Manz et al. (1990), who introduced the concept that a hands-off, self-contained chemical analysis system can be constructed on a miniaturized chip. This was followed by successful demonstration of chip-based fast separation of fluorescent dyes (Manz et al., 1992; Jacobson et al., 1994) and fluorescent labelled amino acids by capillary electrophoresis (CE) (Harrison et al., 1993). Capillary electrophoresis utilizes the different electrophoretic mobilities of various molecular entities (such as proteins) suspended in a carrier fluid to separate them employing an applied electric field. Confining the carrier fluid into narrow microchannels also causes an electroosmotic flow in presence of the applied electric field. This electroosmotic flow transports the fluid along the capillary microchannel, thus obviating the requirement of any additional pumping devices.

1

Loosely defined, systems with dimensions ranging from 1 to 1000 nanometers.

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However, one can additionally utilize pressure gradients to precisely control the flow behavior in microfluidic networks. Following these early developments, there has been an explosive growth in the research and development on microfluidic devices. Many aspects of these developments have been summarized in several excellent reviews and books on the subject (Reyes et al., 2002; Auroux et al., 2002; Beebe et al., 2002; Verpoorte and de Rooij, 2003; Erickson and Li, 2004; Mogensen et al., 2004; Cooper et al., 2004; Darhuber and Troian, 2005; Li, 2004). These reviews elaborately document different aspects of microfluidics, including microfabrication techniques, fundamental aspects of microfluidic flow control, detection and diagnostic systems (such as optical and electrochemical detection techniques), and application in different areas of research such as proteomics, biochemical assays, microfluidic cooling of microchips, etc. A critical aspect in development of microfluidic devices is precise control of fluid transport through the system (Yoshida, 2005). Microfluidic pumping devices constitute the heart of microfluidic chips. Typically, in highly miniaturized microchannels, enormous pressure differences need to be set up to cause a pressure driven flow of a liquid. Consequently, it is more common in such systems to employ electroosmotic flow. Embedding appropriately designed electrodes within a microchannel network can lead to suitable manipulation of the liquid through these channels with very little power requirement. Microchannels also afford a high surface to volume ratio, allowing enhanced heat and mass transfer. The flows are generally in the creeping flow regime (Re ≪ 1). As the size of the system is reduced from microscale to the nanoscale, the size of the confining domain becomes similar to the confined entities, necessitating modification of the boundary conditions of, or the rigor needed in solving, the governing transport equations. As an illustration, consider the flow of an electrolyte solution through a capillary microchannel. For this type of flow, the parameter κa, which represents the ratio of the capillary radius, a, to the thickness of the electric double layer, κ −1 , is of great significance. For a large capillary radius, the parameter κa will have a large value. Most earlier models of electrokinetic flow were developed using the assumption κa ≫ 1. Such an assumption led to considerable simplification of the governing electrochemical transport equations, eventually yielding simple analytical expressions for the electrokinetic velocity of the electrolyte solution in the capillary (see Chapter 8). These simple analytical expressions (for instance, the Helmholtz–Smoluchowski expression) are remarkably accurate for flows in microfluidic channels with radii of the order of 10 to 100 microns. However, when one deals with channel radii of < 1 micron, the parameter κa may no longer be considered large. Consequently, one cannot apply the simple analytical solutions obtained using the assumption of κa ≫ 1. Hence, in nanofluidic applications, the governing electrochemical transport equations are either solved with an alternate set of assumptions (for instance, κa ≪ 1) to obtain another limiting analytical expression for nanoscale capillaries, or with an alternate set of boundary conditions that incorporate the influence of the narrow capillary dimensions in the mathematical model with greater accuracy. Such alterations in the solution methodology do not constitute an alteration of the underlying physics of the transport process. This was demonstrated in the numerical calculations of Chapter 14,

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where the same numerical model based on a set of fundamental governing equations for electrochemical transport worked with great accuracy over the entire range of large and small κa. In this section, we will briefly discuss two applications, which emphasize a slightly different aspect of microfluidics. Commonly, the overarching focus of microfluidics is on miniaturization, leading to applications that are essentially confined to the microscale. While this is definitely a niche, there are few microfluidic applications that directly contribute to the macroscale industrial and analytical operations by enhancing or facilitating them. In these applications, the driving force toward miniaturization was to achieve a technical breakthrough in conventional large scale technologies. In this section, we will briefly discuss two applications that demonstrate how miniaturization can potentially lead to a significant advancement in how we measure properties of large-scale surfaces, or enhance large-scale separation processes. 15.8.1 Measurement of Zeta Potential of Macroscopic Surfaces Measurement of zeta (ζ ) potential of a solid surface immersed in an electrolyte solution is of immense practical significance. The measurement of zeta potential provides insight into the electrical properties of the surface, and how it might affect different processes of industrial relevance. For instance, surface properties of commercial polymeric membranes used for water treatment are routinely probed through measurement of zeta potential to predict their propensity of fouling. In industrial applications dealing with surface coating, it is of paramount importance to have an adequate knowledge of the electrical properties of the surfaces to be coated to ensure proper adhesion of the coating chemicals. Finally, to prevent biofouling on surfaces immersed in an aqueous environment over a long period (such as a ship’s hull), one needs to assess and modify the properties of these surfaces. In all these applications, it is often necessary to measure a representative zeta potential of a very large surface. Virtually any of the electrokinetic transport phenomena discussed in this book can be utilized to provide a measure of the ζ potential of surfaces. These techniques include streaming potential measurement, electroosmotic flow, electrophoresis, and sedimentation potential. Streaming potential measurement in appropriately designed channels have become a commonly used technique for characterizing electrical properties of planar (sheet like) surfaces. There are commercially available instruments for conducting such measurements. Typically, in these devices, one deals with a slitchannel geometry, where two facing walls of the channel are prepared by cutting appropriately sized samples of the material to be tested. These samples fit into a fixed-dimension rectangular channel cell. Cutting solid samples to fit them in the cell constitutes a destructive testing procedure. In this context, development of a streaming potential measurement tool that can conduct in-situ non-destructive measurements of zeta potential of large planar surfaces can be construed as a significant improvement. Walker et al. (2002) conducted a study on the viability of such an electrokinetic cell, dubbed the asymmetric clamping cell, which was originally developed by Anton Paar (Graz, Austria), and is currently marketed by Brookhaven Instruments Corporation. While these authors never explicitly claimed this cell to be a microfluidic system

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in their work, the sub-millimeter channel dimensions used in their study, and the direct applicability of the technique to macroscale surface characterization make it an interesting case study on application of microfluidics to address an important measurement in many large-scale industrial systems. The fundamental concept behind the measurements using the asymmetric clamping cell is to have a probe cell that can be pressed against planar substrates of varied dimensions, followed by taking a quick and facile measurement of the streaming potential. The probe cell consists of a non-conducting grooved poly-methylmethacrylate (PMMA) plate containing several parallel sub-millimeter rectangular channels fabricated using micromachining techniques. The channels are approximately 1 mm wide, 140 µm deep, and 20 mm long. When a planar surface of a test material is pressed against this grooved PMMA manifold, a number of rectangular parallel microchannels are formed. The test material forms one wall of each channel, while the other three walls are formed by the PMMA spacer. The term asymmetric clamping cell stems from the fact that the two opposing walls of the channel develop different zeta potentials when brought in contact with an electrolyte solution. Figure 15.39 schematically depicts the construction of the clamping cell by pressing together a grooved PMMA manifold against a flat substrate, resulting in formation of several parallel microchannels.

Figure 15.39. Schematic representation of (a) a grooved PMMA (poly- methylmethacrylate) spacer and the planar substrate used to construct the asymmetric clamping cell. The assembled structure is shown in (b), depicting the parallel rectangular microchannels formed by pressing the PMMA spacer against the substrate. The top surface of each channel represents the properties of the substrate, while the other surfaces represent the properties of PMMA.

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The analysis of Walker et al. (2002) for the asymmetric clamping cell involved re-deriving the Helmholtz–Smoluchowski formulation for the streaming potential through a two-dimensional slit-microchannel geometry. The two opposing walls of the microchannel have different zeta potentials: one wall represents the zeta potential of PMMA, ζPMMA , while the other represents the zeta potential of the test substrate, ζS . For aqueous electrolyte solutions with monovalent electrolyte concentrations in the range of 1 mM to 100 mM, and a channel half-height, h, of 70 µm, the parameter κh is large (κh ≫ 1), allowing the use of the assumptions inherent in the Helmholtz– Smoluchowski analysis. In this section, we present the generalized derivation of the results obtained by Walker et al. (2002). The geometry of the slit-microchannel used for the analysis is shown in Figure 15.40. The dimensions of the channel are such that h ≪ W ≪ L (Figure 15.40a). Under these conditions, we can assume that the two side walls (edges) of the channel have negligible influence on the electrokinetic flow, and use a two-dimensional slit channel geometry for the analysis as shown in Figure 15.40(b). Furthermore, assuming a long channel, L ≫ h, we can essentially employ the analysis developed for a slit microchannel geometry in Chapter 8. The sole modification of the analysis in this section stems from the use of different zeta potentials on the two opposing channel walls.

Figure 15.40. Geometric representation of the asymmetric clamping cell for the mathematical analysis of streaming potential.

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Under the assumptions stated above, the electric potential distribution in the slitmicrochannel can be represented by the Poisson–Boltzmann equation, linearized assuming the Debye-Hückel limit (low potentials) d 2ψ = κ 2ψ dy 2

(15.38)

where κ is the inverse Debye length. The linearized Poisson–Boltzmann equation can be solved employing the boundary conditions at the two walls of the slit-microchannel, given by ψ = ζS

at y = +h

ψ = ζPMMA

at y = −h

(15.39) (15.40)

The solution of Eq. (15.38) subject to the above boundary conditions provides the electric potential distribution as ψ=

ζS + ζPMMA ζS − ζPMMA cosh(κy) + sinh(κy) 2 cosh(κh) 2 sinh(κh)

(15.41)

The axial momentum balance equation for the electrolyte solution in the slitmicrochannel is given by µ

d 2 ux = −px − ρf Ex dy 2

(15.42)

where µ is the fluid viscosity, ux is the axial velocity of the fluid, px = −∂p/∂x represents the axial pressure gradient, ρf is the volumetric free charge density, given by ρf = −ǫκ 2 ψ

(15.43)

and Ex is the local electric field component in the axial direction. Applying no slip boundary conditions (ux = 0) at the two stationary walls of the slit-microchannel, one can integrate Eq. (15.42) to obtain an expression for the axial velocity profile  
  ǫEx ζS + ζPMMA cosh(κy) y2 −1 1− 2 + h µ 2 cosh(κh)   ζS − ζPMMA sinh(κy) y + − (15.44) 2 sinh(κh) h

p x h2 ux (y) ≡ ux = 2µ

It may be noted that when both the walls have the same zeta potential, the last term in Eq. (15.44) vanishes, yielding the velocity profile derived in Eq. (8.30). Furthermore,

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657

if the electric field in axial direction is vanishingly small, Eq. (15.44) simplifies to the parabolic pressure-driven flow profile ux =

px h2 2µ




1−

y2 h2



(15.45)

Using the expressions for the electric potential and velocity distributions, one can now determine the convective transport current or streaming current per unit channel width along the axial direction, It (see Eq. 8.61). For the present geometry, the streaming current is expressed as It = −ǫ



h

ux

−h

d 2ψ dy = ǫ dy 2



h

−h

dux dψ dy dy dy

(15.46)

where the final expression was obtained employing integration by parts (with ux = 0 at y = ±h). Substituting the velocity profile, Eq. (15.44), and the potential distribution, Eq. (15.41), in Eq. (15.46) one can obtain the streaming current as 2ǫpx h µ

  ζS + ζPMMA tanh(κh) 1− 2 κh


   ǫ 2 κ 2 hEx ζS + ζPMMA 2 tanh(κh) 1 − + µ 2 κh cosh2 (κh)
   ζS − ζPMMA 2 2 coth(κh) 1 + + − 2 κh (κh)2 sinh2 (κh)

It = −




(15.47)

It may be noted that when the zeta potential of the two walls become identical, i.e., ζS = ζPMMA , Eq. (15.47) becomes identical to Eq. (8.67). In the limiting case of κh → ∞, Eq. (15.47) takes the form 2ǫpx h It = − µ




ζS + ζPMMA 2



(15.48)

The conduction current per unit channel width along the axial direction can be expressed as (see Eq. 8.54) 2e2 z2 Dn∞ Ex Ic = kB T



h

−h



1 1+ 2




zeψ kB T

2

∞

dy = σ Ex



h −h



1 1+ 2




zeψ kB T

2

dy

(15.49) where σ ∞ is the bulk electrolyte solution conductivity. Substitution of the electric potential distribution, Eq. (15.41), in Eq. (15.49) followed by evaluation of the

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resulting integral yields



 1 S + PMMA 2 tanh(κh) 1 + Ic = 2σ Ex h 1 + 4 2 κh cosh2 (κh)   

1 S − PMMA 2 coth(κh) (15.50) − + 2 κh sinh2 (κh) ∞

where  = zeζ /kB T . As in the case of streaming current, we note that when ζS = ζPMMA , the conduction current given by Eq. (15.50) becomes identical to the result of Eqs. (8.57) and (8.58). One can further simplify Eq. (15.50) in the Helmholtz– Smoluchowski limit (κh → ∞), yielding Ic = 2σ ∞ Ex h

(15.51)

Equations (15.48) and (15.51) provide the streaming and conduction currents per unit channel width in the Helmholtz–Smoluchowski limit (κh → ∞). One can calculate the total current through a channel of width W by adding these two contributions I = W (It + Ic ) = −

ǫpx hW (ζS + ζPMMA ) + 2σ ∞ Ex hW µ

(15.52)

Setting the total current to zero in Eq. (15.52) yields the required condition for the streaming potential flow, given by ǫ (ζS + ζPMMA ) ǫ Ex = = ζAvg. px µσ ∞ 2 µσ ∞

(15.53)

where (ζS + ζPMMA ) (15.54) 2 With a constant axial electric field (Ex = V /L) and a linear axial pressure gradient (px = −∂p/∂x = p/L), where V and p are the electric potential and pressure differences over a length, L, of the slit-microchannel, respectively, one can rewrite Eq. (15.53) as ǫ V = ζAvg. (15.55) P µσ ∞ It is thus observed that the streaming potential in the Helmholtz–Smoluchowski limit evolves from a simple arithmetic average of the zeta potentials of the two surfaces. Consequently, if the zeta potential of one of the channel walls is known, the corresponding potential on the other wall can be easily determined from the average zeta potential obtained through the streaming potential measurement. Equation (15.55) was employed in the analysis of streaming potential by Walker et al. (2002). The average zeta potential, ζAvg. , was evaluated from the slope of the streaming potential vs. applied pressure difference plots corresponding to a fixed ζAvg. =

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Figure 15.41. Variation of the zeta potential of clean and silanized glass surfaces with electrolyte solution pH measured using the asymmetric clamping cell. The electrolyte contains 0.01 M KCl in all measurements. Experimental data taken from Walker et al. (2002).

electrolyte concentration and pH. Using a known value of the PMMA zeta potential under the same experimental conditions, the unknown zeta potential of the substrate, ζS , was then evaluated using Eq. (15.54). Figure 15.41 depicts representative results from the analysis of Walker et al. (2002), where the zeta potential of clean quartz glass and a silanized sample of glass coated with aminoethyl-aminomethyl-phenethyltrimethoxysilane (AAPhenol) were measured in an aqueous solution of 0.01 M KCl over a wide range of pH values. It is evident from the figure that the silanized glass has a distinctly different and more positive surface compared to the native glass. The point of zero charge (corresponding to the pH where the zeta potential is zero) was found to be shifted from about 2 for quartz glass to about 6.5 for the silanized glass. These results clearly indicate the ability of the asymmetric clamping cell to provide reliable estimates of the zeta potential of differently charged substrates using facile measurements of the streaming potential. The striking feature of these measurements is that the small self-contained electrokinetic cell with the grooved polymeric spacer was simply clamped to large pieces of the substrates without having to “process” the surfaces to be tested. One can apply the principles of the asymmetric clamping cell to construct even narrower channels, thus reducing the overall size of the device. However, one should note that the pressure gradients required to drive the fluid through narrower microchannels will increase considerably in such situations. As well, the general forms of the governing equations applicable for smaller channel heights (small κh) need to be used in such situations. 15.8.2 AC Electrokinetics: Application in Membrane Filtration In this book, we solely focussed on electrokinetic phenomena in a direct current (DC) electric field. The application of alternating current (AC) fields to cause electrokinetic

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effects is an equally well-developed subject, with far reaching applications in microfluidics and biology (Pohl, 1978; Jones, 1995; Hughes, 2003). The alternating electric field, as the term alternating suggests, arises from periodic (oscillatory) variations in the magnitude and direction of the electric field with time. When a polarizable multicomponent fluid is subjected to such an alternating electric field, one observes additional electrokinetic phenomena that are dependent on the frequency of the AC field. Some of the key features of electrokinetic phenomena observed in an AC field are: 1. The dielectric permittivities of materials subjected to an AC field become a complex function of the frequency of the imposed field. The complex dielectric permittivity, ǫ ∗ , of a material is expressed as ǫ∗ = ǫ − i

σ ω

(15.56)

√ where ǫ is the real part of the permittivity, i = −1 is the imaginary number, σ is the electric conductivity of the material, and ω is the frequency of the AC signal.2 2. A charged particle placed in a spatially uniform AC electric field will experience no Coulomb force, since the time average of the electric field will be zero. In other words, averaged over time, the net Coulomb force is given by Fp  = Qp < E >= 0

(15.57)

where · · ·  denotes a time averaged quantity, Fp is the Coulomb force, Qp is the charge on the particle, and E is the electric field. This is an important concept in AC electrokinetics, since at high frequencies, say ω → 1 MHz (megaHertz), even an electrolyte solution with a substantial salt concentration will become virtually non-conducting. Furthermore, there will be no electrode polarization (formation of electric double layers) in presence of such high frequency AC fields. 3. A particle with a complex dielectric permittivity, ǫp∗ , placed in a continuous medium with a complex dielectric permittivity, ǫm∗ , will experience a dielectric polarization force in a nonuniform AC field, which is proportional to ∇(E · E). This force is called the dielectrophoretic force (Pohl, 1978; Jones, 1995). An approximate expression for the time averaged dielectrophoretic force, < FDEP >, on a spherical particle of radius ap is FDEP  = 2π ap3 ǫm Re[fCM ] ∇(E · E)

(15.58)

where ǫm is the real part of the permittivity of the suspending medium and Re[fCM ] is the real part of a complex number, fCM , known as the Clausius– Mossotti factor. The Clausius–Mossotti factor gives the frequency dependence 2

When we refer to “AC signal”, we imply an applied voltage that varies with time between a positive and negative peak value, usually sinusoidally.

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661

Figure 15.42. Variation of the real part of the Clausius–Mossotti factor with frequency of the AC signal for polystyrene latex particles in water. The crossover frequency demarcates the two regimes of dielectrophoresis. At lower frequencies, the system will exhibit positive dielectrophoresis, while at higher frequencies, the same system will exhibit negative dielectrophoresis.

of the polarization effect, and is expressed by fCM =

ǫp∗ − ǫm∗

ǫp∗ + 2ǫm∗

(15.59)

The real part of the Clausius–Mossotti factor changes with the frequency of the AC signal, and varies between the limits of +1 to −0.5. The frequency dependent real part of the Clausius–Mossotti factor for polystyrene latex particles suspended in water is depicted in Figure 15.42. When the factor is positive, the dielectrophoretic force acts along the direction of the positive gradient of the electric field intensity (toward regions of high field intensities). In contrast, when the factor is negative, the force acts along the negative gradient of the field intensity (toward regions of low field intensities). In Eq. (15.58) the term E · E represents the time averaged electric field intensity. Note that in an AC system, although the time averaged electric field, E is zero, the corresponding time averaged field intensity, E · E =< E 2 > is a finite quantity. Migration of particles along the positive electric field intensity gradient is termed as positive dielectrophoresis, while the opposite phenomenon is referred to as negative dielectrophoresis (Hughes, 2003). It is important to note that the dielectrophoretic forces are felt only when a spatial gradient of the electric field intensity exists. These forces are absent when the electric field is spatially uniform. Another important aspect of dielectrophoretic forces is that one requires an enormous field (generally in the range of 106 V/m) to generate appreciable forces. In macroscopic systems, when

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the electrodes are large, one requires very large voltages to obtain measurable dielectrophoretic effects. However, if the electrode dimensions and spacings are decreased to the micrometer range, one can easily generate local electric fields of the order of 106 V/m by applying potentials of 1 to 10 V. Thus miniaturization is an important aspect of dielectrophoretic manipulation. AC electrokinetic techniques based on dielectrophoresis have been utilized for manipulation, separation, and characterization of colloidal scale entities for several decades, although their application received heightened attention during the past decade (Green and Morgan, 1997; Hughes, 2000; Gascoyne and Vykoukal, 2002). The application of dielectrophoresis is particularly prevalent in the field of biology, where these forces are used to trap and sort different types of cells, and fractionate components of a cellular mixture (Pethig and Markx, 1997). Application of dielectrophoresis in flocculation of colloidal entities is also known (Eow et al., 2001). In this section, we will briefly describe a novel concept of combining dielectrophoresis with tangential flow membrane filtration, which can be used as a convenient mechanism for mitigation of particulate fouling of membranes. Once again, the rationale for describing such an application is to demonstrate that by judiciously applying a microscale technique, one might affect an advancement in a conventional large-scale technology. Membrane filtration processes such as microfiltration and ultrafiltration operate in commercial scale applications in a continuous mode. In these applications, the feed suspension is circulated under high pressure through a membrane module. The porous wall of the membrane module is made of polymeric or ceramic membranes, which selectively allows the solvent to permeate due to the transmembrane pressure difference, and retain the solutes. In most applications involving microfiltration, the feed suspension contains colloidal particles or macromolecules. The retained solutes (colloidal particles) accumulate near the membrane surface, and eventually block the membrane pores. This phenomenon is termed as particulate fouling of the membrane. Fouling results in reduction of the solvent permeation rate through the membrane, as well as, over a long term operation, causes irreversible damage to the membrane, necessitating its replacement. Numerous techniques for mitigation of membrane fouling have been devised, although no viable technique for in situ fouling prevention is currently available (Molla and Bhattacharjee, 2005). In this context, Molla et al. (2005) and Molla and Bhattacharjee (2005) proposed the use of dielectrophoresis on a parallel electrode array placed over a membrane as a possible mechanism for abatement of membrane fouling. The underlying concept of their approach is to embed a microfabricated parallel electrode array on a membrane. A photograph of a sample microfabricated parallel electrode array on a glass substrate is shown in Figure 15.43. Each parallel electrode in this array is about 50 µm wide, with a gap of 50 µm between consecutive electrodes. The overall array consists of 200 electrodes. The electrodes are arranged as intercalated combs, with consecutive electrodes connected to a different busbar. The two busbars are powered by a 180◦ phase shifted AC power supply with an adjustable root mean square (RMS) voltage and frequency setting. The 180◦ phase shift implies that at a given

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Figure 15.43. Photograph of a microfabricated gold electrode array on a glass substrate. (a) The full array with 200 electrodes. (b) Magnified view of a few electrodes within the rectangle in part (a) showing the intercalated comb structure.

instant, two consecutive electrodes will have their surface potentials set to +V and −V . Application of such a potential to the parallel electrode array ensures that we have stationary wave dielectrophoresis in the system. In other words, the electric field intensity gradients, ∇(E · E), do not move spatially with time in this case. Molla and Bhattacharjee (2005) conducted numerical simulations of colloidal particle trajectories near a membrane during tangential flow filtration in presence of dielectrophoretic forces. The two dimensional rectangular channel geometry used in their simulations is depicted in Figure 15.44. They numerically evaluated the electric potential distribution, the electric field intensities, and the time averaged dielectrophoretic forces acting on individual colloidal particles at different distances from the membrane surface. They used the finite element technique to solve the Laplace equation for the electric potential distribution in the fluid. The boundary conditions at the electrode surfaces were set to constant surface potential, although the potential was varied sinusoidally with time emulating the AC signal. From the potential and electric field distributions, the electric field intensity gradient components parallel and perpendicular to the membrane surface (along the x and y directions in Figure 15.44, respectively) were determined. These were substituted in Eq. (15.58) to calculate the components of the time averaged dielectrophoretic force on a particle along these directions. In their study, Molla and Bhattacharjee (2005) also obtained a numerical solution of the Navier-Stokes equation for the fluid flow in presence of an intermittently permeable wall channel. It should be noted that placing impermeable electrodes covering part of the membrane decreases the membrane permeability, and the product water flux. Consequently, one needs to appropriately factor in this reduction of flux to study

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Figure 15.44. Schematic diagram of the geometry used for modeling the dielectrophoretic crossflow membrane filtration system. The tangential velocity in the channel is represented through the parabolic distribution. The gap between the electrodes is assumed to be the same as the electrode width. All dimensions are scaled with respect to the electrode width, W .

the benefits of using dielectrophoretic membrane filtration as a mechanism for reducing fouling. This necessitates a detailed accounting of the influence of intermittently permeable membrane on the solvent flux and the particle hydrodynamics. Utilizing the dielectrophoretic forces and the fluid velocity field in a particle trajectory analysis as outlined in Chapter 13, Molla and Bhattacharjee (2005) obtained representative particle trajectories in presence of attractive and repulsive dielectrophoretic forces. They studied a system of polystyrene latex particles in water undergoing steady-state crossflow filtration in a two-dimensional rectangular channel. In all simulations, the normal fluid velocity at the membrane surface (termed suction velocity) was set to 10 µm/s. The average tangential velocity in the channel was 1 mm/s. Figure 15.45 depicts some of the representative particle trajectories obtained under different conditions. The trajectory labeled “uniform suction” represents the motion of the particle in absence of any dielectrophoresis on an unmodified membrane. Since there are no electrodes blocking the membrane surface, this situation corresponds to a standard particle trajectory in conventional membrane filtration. Notably, since the particle trajectory terminates at the membrane surface, this condition implies that the particle will deposit on the membrane and foul it. When one places the electrodes on the membrane, thus making it intermittently permeable, one obtains the trajectory labelled “intermittent suction”. It is evident from these two trajectories that partially blocking the membrane surface slows down the approach of the particle toward the membrane, thus slightly reducing the fouling tendency. The two trajectories labelled positive and negative dielectrophoresis in Figure 15.45 were obtained by applying the dielectrophoretic force. The positive

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665

Figure 15.45. Comparison of four trajectories of a 1 µm radius polystyrene latex particle obtained under different conditions. All trajectories were calculated assuming the same fluid velocities. The average tangential and normal permeation velocity components were 10−3 m/s and 10−6 m/s, respectively. The initial position of the particle was fixed at a height of 50 µm from the electrode surface. All lengths are scaled with respect to the electrode height W = 50 µm. (Adapted from Molla and Bhattacharjee, 2005.)

dielectrophoresis occurs in the system at a low frequency of the applied AC signal, while negative dielectrophoresis occurs at a high frequency. This is evident from the frequency dependence of the Clausius–Mossotti function depicted in Figure 15.42. The remarkably large magnitude and range of the dielectrophoretic forces is evident from Figure 15.45, since a particle at a distance of 50 µm from the electrode surface can evidently feel this force, and migrate toward or away from the membrane under the influence of this force. The magnitude and range of the dielectrophoretic forces are much larger compared to those of the electric double layer and van der Waals forces. It is evident from the negative dielectrophoretic trajectory that one can use these forces to repel the particles from the membrane, levitating them further away from the membrane. Such a force is independent of the presence of charge on the colloidal particles. The technique can thus serve as a unique mechanism for in-situ prevention of membrane fouling without having to stop the filtration operation for cleaning. In presence of dielectrophoretic levitation, since the fluid immediately adjacent to the membrane will be essentially devoid of particles, there will be no reduction in permeate flux. In fact, the membrane will operate as if it is “filtering” pure solvent, thus maximizing the solvent flux through its pores. The above example of a microfluidic application based on AC electrokinetics to a conventional large-scale separation once again demonstrates the immense potential of microscale phenomena in providing significant breakthroughs in conventional industrial applications. The example also illustrates that the principles underlying AC electrokinetics are essentially embedded in the general framework of electrostatics

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and electrokinetics dealt with in this book. The interesting fact about the underlying electrostatics in both AC and DC electrokinetics is that the same Poisson and Laplace equations can be used over a wide range of frequencies. Dynamic effects, such as electromagnetic coupling become important only when extremely high frequencies are used (ω > 10 MHz). Thus, an in depth knowledge of electrostatics, and some additional insight regarding frequency dependence of dielectric polarizability and permittivity of materials allow one to take the theories of DC electrokinetics to the realm of AC electrokinetics.

15.9

NOMENCLATURE

a ap BGL c(x) c1 c2 c¯ ci cf c± (r, x) Csc D∞ e Ex E fCM F Fp FDEP h i Js kB L n px Pe P e′ Qp , q r R Re

capillary tube radius, m particle radius, m blood glucose level ionic concentration at an axial distance x along the capillary axis, mol/m3 molar feed salt concentration, mol/m3 molar outlet salt concentration, mol/m3 dimensionless ionic concentration initial oil droplet concentration final oil droplet concentration ionic concentration, mol/m3 electric capacitance of the stratum corneum, F diffusion coefficient at infinite dilution, m2 /s elementary charge, C axial component of electric field, V/m electric field vector, V/m Clausius–Mossotti factor Faraday constant, C/mol Coulomb force on a charged particle, N dielectrophoretic force, N half-height of slit-microchannel, m √ imaginary number, −1 total flux of dissociated salt, mol/s Boltzmann constant, J/K capillary tube length, m particle concentration m−3 axial pressure gradient in slit microchannel, Pa/m rescaled particle Peclet number particle Peclet number charge on a particle, C radial coordinate, m scaled radial coordinate Reynolds number

15.9

Rg Rr Rsc Rvs T ux (r) V x z

NOMENCLATURE

universal gas constant, J mol−1 K−1 salt rejection coefficient of capillary electrical resistance of the stratum corneum,  electrical resistance of viable skin,  absolute temperature, K local axial velocity, m/s average fluid velocity, m/s axial coordinate, m valency

Greek Symbols αm αp γ˙ V ǫ ǫ ∗ , ǫm∗ , ǫp∗ ηT [η] κ λ λ1 λD µ µ0 µ∞ µc µr µs φ(x)
(r, x) σ σ∞ ζAvg. ζPMMA ζS ψ(r, x) ¯ x) ψ(r, ψ¯ w S , PMMA ρf τ τr ω

maximum packing volume fraction of particles volume fraction of dispersed phase shear rate, s−1 streaming potential, V dielectric permittivity of medium, C/(Vm) complex dielectric permittivities flotation efficiency intrinsic viscosity inverse Debye length, m−1 dimensionless Debye length parameter dimensionless Debye length at capillary inlet Debye length parameter, m−1 fluid viscosity, Pa.s limiting suspension viscosity at γ˙ → 0 limiting suspension viscosity at γ˙ → ∞ continuous phase viscosity, Pa.s relative viscosity of a colloidal suspension suspension viscosity, Pa.s induced potential, V total potential, V conductivity of material, S/m bulk electrolyte solution conductivity, S/m average zeta potential, V zeta potential of PMMA, V zeta potential of test substrate, V potential due to electric double layer, V dimensionless potential due to electric double layer dimensionless capillary wall potential scaled zeta potentials, (zeζ /kB T ) free charge density, C/m3 shear stress, N/m2 reduced shear stress frequency of an AC voltage (signal), s−1 or Hz

667

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15.10

REFERENCES

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INDEX

Additivity rule, colloidal particle deposition, spherical collectors, 496–497 Adhesion parameter: colloidal particle deposition, flux equation, dimensionless groups, 527 oil sand components, bitumen extraction, 641 fine solids interaction, 648–650 Air–water interface: colloidal particles, 20 oil droplet/fine particle flotation electrokinetics, 622–624 Algebraic equations, numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, 550 Alternating current (AC) electrokinetics, membrane filtration, 659–666 Anion exchange membrane, electrodialysis, 27 Anomalous surface conduction, microchannel flow, 274 Aqueous electrolyte solutions: conducting materials, 55–56 London–van der Waals forces, DLVO theory, 418–420

multicomponent systems, molar electric conductivity, 194–198 planar electric double layer, Debye–Hückel approximation, 117–122 transport equations, 184 Asymmetric electrolytes. See also Symmetric electrolytes curved electric double layer, spherical geometry, 132–136 planar electric double layer, Debye–Hückel approximation, 121–122 Atomic force microscopy (AFM), oil sand components, bitumen extraction, 640–641 fine solids interaction, 648–650 silica interactions, 644–645 Attraction forces: colloidal particle deposition, trajectory equation, dimensionless groups, 528 shear coagulation, 456–462

Electrokinetic and Colloid Transport Phenomena, by Jacob H. Masliyah and Subir Bhattacharjee Copyright © 2006 John Wiley & Sons, Inc.

673

674

INDEX

Axial symmetry: dilute suspensions, Ohshima’s cell model, 372 numerical simulation, electrokinetic phenomena: cylindrical capillary model, 562–567 electroosmotic flow, 581–587 Poisson–Boltzmann equation, 543–546 Band width variance, solute dispersion, microchannel flow, convection-diffusion, 281–283 Barometric equation, electric double layers, interfacial charges, Boltzmann distribution, 110–111 Batchelor equations, colloidal suspensions rheology, hard sphere model, 627–628 Bessel function: circular charged capillary, electroosmotic flow, 255–257 double layers, 258–261 Binary electrolyte solution, multicomponent systems, 199–201 Bispherical coordinates, numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation validation, 554–557 Bitumen extraction, oil sands, electrokinetics, 632–650 atomic force microscope technique, 640–641 bitumen-fine solids interaction, 645–650 bitumen-silica interaction, 641–645 zeta potential: distribution measurements, 639–640 oil sand components, 635–638 “Black-box” code, numerical simulation, electrokinetics, Poisson–Boltzmann equation, finite element formulation, 546–547 Blocking process, colloidal particle deposition: long-term behavior, 522–523 porous media transport models, 524–526 Blood glucose levels, iontophoretic electrokinetics, 621 Boltzmann distribution: circular charged capillary, electroosmotic flow, 255–257 dilute suspensions, Ohshima’s cell model, perturbations, 375–376

electric double layers, interfacial charges, 109–111 electrophoretic mobility: arbitrary Debye lengths, Henry’s solution, 312–322 Ohshima cell model, 341–344 London–van der Waals forces: DLVO verification, 414–415 Schulze–Hardy Rule, 410–412 multicomponent systems, 201–203 numerical simulation, electrokinetics: capillary microchannel, transient electrolyte transport, 579–581 electrophoretic mobility, 588–590 particle coagulation, Brownian motion, 432 planar electric double layer: Debye–Hückel approximation, 119–122 Gouy–Chapman analysis, 112–114 ionic concentrations, 125–128 salt rejection electrokinetics, porous media/membranes, 614–617 slit charge microchannels, electroosmotic flow, 232–235 electric current density, 246–250 surface ionization models, 167–169 Bond number, air-water interface, 21 Boundary conditions: Brownian coagulation, Smoluchowski solution, without field force, 434–437 circular charged capillary, electroosmotic flow, 255–257 colloidal particle deposition, Eulerian approach, stagnation flow, 488–490 concentrated suspensions, sedimentation potential, 382–386 curved electric double layer: cylindrical coordinates, 136–138 spherical geometry, 130–136 dilute suspensions, Ohshima’s cell model, 373–374 dissimilar surfaces, overlapping planar layers, surface charge density, 149–151 electrophoretic mobility: arbitrary Debye lengths: Henry’s solution, 313–322 perturbation approach, 310–311 hydrodynamic cell models, 330–332 mobility, Levine–Neale cell model, 333–340

INDEX

Ohshima cell model, 341–344 Shilov–Zharkikh cell model, 346–352 single charged sphere, 299–303 Debye length limits, 303–308 relaxation, 301–302 retardation, 301 surface conductance, 302–303 electrostatic equations, 62–68 conducting sphere, external electric field, 91–97 spherical dielectrics, 84–86 two-dimensional dielectric slab, external electric field, 78–80 macroscopic surfaces, zeta potential measurement, 656–659 numerical simulation, electrokinetic phenomena: capillary microchannel: electrolyte transport, transient analysis, 577–581 streaming potential, 570–572 computer tools and methods, 539–541 electroosmotic flow, 582–587 electrophoretic mobility, 592–596 Henry’s function, 597 Poisson–Boltzmann equation, 544–546 charged capillary particles, EDL interaction, 558–559 validation of results, 554–557 particle coagulation, Brownian motion, 430–432 planar electric double layer, charged planar surfaces, electrostatic interaction, 140–144 salt rejection electrokinetics, porous media/membranes, 615–617 shear coagulation, 460–462 slit charge microchannels, electroosmotic flow, 233–235 electric potential, 233–235 flow velocity, 235–238 Boundary element techniques, numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, 543–559 Bound charge, dielectric materials polarization, 53–56 Brinkman cell model, colloidal particle deposition, 520–521 Brownian motion. See also London–van der Waals forces coagulation dynamics, 428–429

675

colloidal particle deposition: classical convection-diffusion transport, 471 Eulerian approach: spherical collector, 478–482 stagnation flow, 482–490 Lagrangian approach, 470–471, 502–509 colloidal particles, 17–18 historical background, 28–29 colloidal suspensions rheology, 625–632 hard sphere model, 626–628 electrical potential distribution, electric double layers, interfacial charges, 108–109 particle coagulation, 429–432 basic principles, 428 shear coagulation in absence of, 448–451 Smoluchowski solution: field force effects, 437–448 without field force, 434–437 Broyden search, numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, 550 Capacitor charging, dielectric materials polarization, 54–56 Capillary electrophoresis, microfluidics, 651–653 Capillary microchannel. See also Cylindrical capillary model numerical simulation, electrokinetic phenomena: electrolyte transport, transient analysis, 577–581 electroosmotic flow velocity profiles, 584–587 streaming potential, 570–577 boundary conditions, 570–572 numerical vs. analytical results, 572–577 Capture efficiency, colloidal particle deposition: interception, 471–475 Lagrangian approach, 506–509 Sherwood number and, 509–512 spherical collectors, 495–497 Carrique’s expression, electrophoretic mobility, Shilov–Zharkikh cell model, 351–352 Cartesian coordinates: basic components, 7–8

676

INDEX

Cartesian coordinates (Continued) Brownian coagulation, field force effects, 446–448 electrostatics, boundary conditions, 66–68 equations of motion, 215 linear dielectric, Maxwell stress, 70–73 numerical simulation, electrokinetics, Poisson–Boltzmann equation, 546–547 planar electric double layer, Debye–Hückel approximation, 118–122 slit charge microchannels, electroosmotic flow, 232–235 tensor operations, 11 Cationic exchange membrane: electrodialysis, 27 oil sand components, bitumen extraction, 636–68 Cell models electrophoretic mobility: conductivity, 344–346 hydrodynamic cell models, 328–332 hydrodynamics, 328–332 Levine–Neale cell model, 333–340 Ohshima cell model, 340–344 prediction accuracy, 352–353 Shilov–Zharkikh cell model, 346–352 Charge conservation: electrostatics, 33–36 multicomponent systems, 198–199 Charged capillary particles, numerical simulation, electrokinetic phenomena: electroosmotic flow velocity profiles, 584–587 Poisson–Boltzmann equation, EDL interaction force, 557–559 Charged discs, dielectric medium, electrostatic properties, 95–97 Charge density: dielectric electrostatics, 57–62 electric field strength, 37–38 electrostatics: boundary conditions, 65–68 two-dimensional dielectric slab, external electric field, 80–81 Gouy–Chapman model: arbitrary electrolyte, 171 symmetric electrolytes, 172 Maxwell’s electromagnetism equations, 73–74

numerical simulation, electrokinetic phenomena: electrophoretic mobility, 591–596 Henry’s function, 597 Poisson–Boltzmann equation, 545–546 Charged interfaces. See Interfacial charge Charged parallel plates: electrostatic potential energy, 152–155 planar electric double layer, electrostatic interaction, 144–146 Charged planar surfaces, planar electric double layer, electrostatic interaction, 138–144 Charged spherical shell, electrostatics: electric field strength, 41–43 electric potential, 49–50 Charge quantization, electrostatics, 33–36 Chemical heterogeneity, colloidal particle deposition, 523–524 porous media transport, 526 Circular charged capillary, electroosmotic flow, 253–268 current flow, 262–265 electroviscous effect, 266–268 Helmholtz-Smoluchowski equation, thin double layers, 257 streaming potential analysis, 265–266 thick double layers, 257–261 Circular cylindrical capillary, electrophoretic mobility, 354–356 Clausius–Mossotti factor, alternating current electrokinetics, 660–666 Closed slit microchannel, electroosmosis, 240–243 Coagulation, of particles. See Particle coagulation Cohesive work, London–van der Waals forces, Hamaker constant, 401–403 Coion distribution, planar electric double layer, 126–128 Collector bed heterogeneity, colloidal particle deposition, 523–524 Collision frequency: Brownian coagulation, field force effects, 438–448 coagulation dynamics, 428–429 colloidal particle deposition: interception, 474–475 Lagrangian approach, 497–509, 503–509 particle coagulation, 432–433 shear coagulation, 449–451, 456–462

INDEX

Colloidal particles: coagulation dynamics, 428–429 deposition: basic principles, 469–471 Brownian diffusion, classical convection-diffusion transport, 471 deposition: experimental verification, 512–521 particle collisions, spherical collector, 497–509 Sherwood number, 509–512 dimensionless groups, 527–528 Eulerian approach, 477–497 spherical collector: Brownian deposition with external forces, 490–497 Brownian deposition without external forces, 478–482 stagnation flow, Brownian diffusion with external forces, 482–490 flux equation, 527 inertial deposition, 475–476 interception, 471–475 Lagrangian approach, 497–509 porous media, 521–524 chemical heterogeneity, 523–524 hydrodynamic dispersion, 521–522 long-term behavior, 522–523 release mechanisms, 523 transport models, 524–526 spherical collector, 497–509 trajectory equation, 527–528 electrokinetic phenomena, 29–30 electrophoretic mobility, hydrodynamic cell models, 328–332 examples of, 15 historical summary, 27–29 hydrodynamics, transport equations, 205–212 London–van der Waals forces: dispersion, bodies in vacuum, 391–393 DLVO theory, 406–409 limitations, 416–420 nomenclature, 30–31 phenomena, 16–21 physical state, 13–16 preparation of, 23–26 condensation methods, 24–26 dispersion methods, 23–24 sols purification, 26–27 stabilization, 21–23

677

Colloidal suspension rheology, 625–632 electroviscous effects, 628–632 hard sphere model, 626–628 historical background, 625–626 Computational geometry, numerical simulation, electrokinetic phenomena, cylindrical capillary model, 562–567 Computer-based simulation, electrokinetics phenomena, 538–541 Concentrate channel, sol purification, 27 Concentrated suspensions: electrophoretic mobility, 327–353 cell model prediction accuracy, 352–353 conductivity, 344–346 hydrodynamic cell models, 328–332 Levine–Neale cell model, 333–340 Ohshima cell model, 340–344 Shilov–Zharkikh cell model, 346–352 sedimentation potential, 381–386 Condensation methods, colloidal system, 24–27 Conducting sphere, external electric field, electrostatic forces, 91–97 electric potential and field strength, 91–92 Maxwell electrostatic stress, 94–95 surface charge density, 92–93 Conduction current: numerical simulation, electrokinetic phenomena, capillary microchannel streaming potential, 574–577 slit charge microchannels, electroosmotic flow, 245–250 Conductivity: circular charged capillary, electroosmotic flow, current flow, 263–265 electrophoresis, suspension conductivity, 344–346 multicomponent systems, current density, 191–198 Conductors, electrostatic classification, 51–56 Conjugate driving force, electrokinetics, Onsager reciprocity relationships, 224–226 Conservation principles, Maxwell’s electromagnetism equations, 73–74 Constant fluid density, continuity equation, 215

678

INDEX

Constant surface charge/potential: electrophoresis, velocity equations, 323–324 numerical simulation, electrokinetic phenomena: electrophoretic mobility, 602–605 Poisson–Boltzmann equation, 544–546 Continuity equation: constant fluid density, 215 dilute suspensions, Ohshima’s cell model, 371–372 numerical simulation, electrophoretic mobility, 587–590 solute dispersion, microchannel flow, convection-diffusion, 279–283 summary of, 215 Convection–diffusion equation: colloidal particle deposition: Brownian diffusion, 471 Eulerian approach, 470 stagnation flow, 486–490 porous media transport models, 524–526 spherical collectors, 495–497 multicomponent systems, binary electrolyte solutions, 199–201 single-component system, 181 solute dispersion, microchannel flow, 278–283 non-uniform dispersion, Taylor–Aris theory, 282–283 uniform flow dispersion, 280–282 Convection–diffusion–migration equation, multicomponent systems, 185–191 Convective transport: electrophoresis, relaxation effects, 324–327 slit charge microchannels, electroosmotic flow, 245–250 Conversion factors, non-SI units, 4 Correction functions: colloidal particle deposition, Eulerian approach, 478 hydrodynamics of colloidal systems, 205–211 Cost analysis, numerical simulation, electrokinetic phenomena, 604–605 Coulomb’s law: alternating current electrokinetics, 660–666 dielectric materials polarization, 53–56 electrostatics, 34

dielectric materials, 60–62 Counterion concentrations: London–van der Waals forces: DLVO verification, 413–415 Schulze–Hardy Rule, 410–412 planar electric double layer, 126–128 surface potentials and, 128–130 Creeping flow problem, shear coagulation, hydrodynamic and field forces, 451–462 Critical flocculation concentration (CFC), London–van der Waals forces, Schulze–Hardy Rule, 409–412 Cross effects, electrokinetics, Onsager reciprocity relationships, 225–226 Crystalline structures, surface ions, electric double layers: charged surfaces, 107 differential dissolution, 106–107 Cubic subdivision, surface molecules, 19 Current density: circular charged capillary, electroosmotic flow, 262–265 conservation equation, 212 electrolyte solution, 214 multicomponent systems, 191–198 binary electrolyte solutions, 199–201 slit charge microchannels, electroosmotic flow, 244–250 Curved electric double layer, 130–138 Debye–Hückel approximation: cylindrical geometry, 136–138 spherical geometry, 130–136 electrostatic interactions, 155–165 approximate solutions, 164–165 Derjaguin approximation, 157–162 linear superposition approximation, 162–164 Curve-fit expressions, colloidal particle deposition, Eulerian approach, stagnation flow, 484–490 Cylindrical capillary model: numerical simulation, electrokinetic phenomena: computer tools and methods, 540–541 electrolyte flow, 559–587 computational geometry and governing equations, 562–567 electroosmosis, axial pressure and potential gradients, 581–587 mesh generation and numerical solution, 567–570

INDEX

microchannel streaming potential, 570–577 transient analysis, microchannel transport, 577–581 electrophoretic mobility, 588–590 Poisson–Boltzmann equation, 542–546 EDL interaction force, 557–559 salt rejection electrokinetics, porous media and membranes, 613–617 Cylindrical coordinates: basic components, 8 circular charged capillary, electroosmotic flow, 254–257 colloidal particle deposition: Eulerian approach, stagnation flow, 485–490 inertial deposition, 475–476 Lagrangian approach, 498–509 curved electric double layer, Debye–Hückel approximation, 136–138 electrophoresis: circular cylinders, electric field, 354–356 Henry’s function, 321–322 equations of motion, 215 London–van der Waals forces, Hamaker’s approach, 394–403 numerical simulation, electrokinetic phenomena: electrophoretic mobility, 590–596 Poisson–Boltzmann equation, 543–559 finite element formulation, 546–547 Cylindrical volume, electrostatics, boundary conditions, 63–68 Darcy’s equation, electrokinetics, Onsager reciprocity relationships, 224–226 Debye–Hückel approximation: circular charged capillary, electroosmotic flow, 255–257 curved electric double layer: cylindrical geometry, 136–138 electrostatic forces, 157 linear superposition approximation, 164 spherical geometry, 130–136 electrophoresis: arbitrary Debye lengths, perturbation approach, 311

679

circular cylinders, electric field, 354–356 mobility, Levine–Neale cell model, 334–340 single charged sphere, 303–306 electrostatic potential energy, 153–155 Gouy–Chapman model, 172 London–van der Waals forces, DLVO theory, 406–409 macroscopic surfaces, zeta potential measurement, 656–659 planar electric double layer: basic principles, 116–122 counterion analysis, surface potentials, 129–130 Gouy–Chapman analysis, 114 ionic concentrations, 126–128 slit charge microchannels, electroosmotic flow, 233–235 electric current, 244–250 Debye length: colloidal particle deposition: Eulerian approach, stagnation flow, 487–490 Lagrangian approach, 507–509 colloidal suspensions rheology, electroviscous effects, 628–632 concentrated suspensions, sedimentation potential, 384–386 dilute suspensions, Ohshima’s cell model, perturbations, 375–376 electrophoresis: arbitrary length solutions, 308–327 alternate velocities, 322–324 Henry’s solution, 311–322 perturbation approach, 309–311 relaxation effects, 324–327 relaxation effects, 325–327 single charged sphere: greater than 1, 306–308 less than 1, 303–306 London–van der Waals forces, DLVO theory, 406–409 macroscopic surfaces, zeta potential measurement, 656–659 numerical simulation, electrokinetics: cylindrical capillary model, electrolyte flow, 565–567 electrophoretic mobility, 594–596 planar electric double layer: charged planar surfaces, electrostatic interaction, 143–144

680

INDEX

Debye length (Continued) Debye–Hückel approximation, 117–122 Gouy–Chapman analysis, 114–116 salt rejection electrokinetics, porous media and membranes, 614–617 slit charge microchannels, electroosmosis, 233–235 electric current density, 247–250 Degrees of freedom (DOF), numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, 550 Depletion forces, DLVO theory, 419–420 Deposition: colloidal particles: basic principles, 469–471 Brownian diffusion, classical convection-diffusion transport, 471 dimensionless groups, 527–528 Eulerian approach, 477–497 spherical collector: Brownian deposition with external forces, 490–497 Brownian deposition without external forces, 478–482 stagnation flow, Brownian diffusion with external forces, 482–490 experimental variations, 512–521 flux equation, 527 inertial deposition, 475–476 interception, 471–475 Lagrangian approach, 497–509 particle collisions, spherical collector, 497–509 porous media, 521–524 chemical heterogeneity, 523–524 hydrodynamic dispersion, 521–522 long-term behavior, 522–523 release mechanisms, 523 transport models, 524–526 Sherwood number, 509–512 trajectory equation, 527–528 oil droplet/fine particle flotation electrokinetics, 622–624 particle coagulation, 427–428 Derived quantities, SI units, 2 Derjaguin approximation: curved electric double layer, electrostatic forces, 157–162 London–van der Waals forces, Hamaker constant, 398–400

numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation validation, 554–557 Derjaguin–Landau–Verwey–Overbeek theory. See DLVO theory Deybe length: colloidal suspensions rheology, 625–632 numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, 543–546 Dialysate channel, sol purification, 27 Dialysis, sol purification, 26–27 Dielectric constants: electrostatics of liquids and solids, 61–62 water temperature variation, 62 Dielectric materials: electrostatics: basic principles, 56–62 charged and parallel discs, 95–97 classification, 51–56 sphere, external electric field, 83–90 electric potential and field strength, 84–86 Maxwell electrostatic stress, dielectric sphere, 87–90 polarization surface charge density, 86–87 two-dimensional dielectric slab, external electric field, 77–83 electric potential and field strength, 78–80 Maxwell electrostatic stress, 81–83 polarization surface charge density, 80–81 intervening medium, London–van der Waals forces, 403–406 linear dielectrics, Maxwell stress tensor, 68–73 Dielectrophoretic force, alternating current electrokinetics, 660–666 Differential dissolution, surface ions, electric double layers, 106–107 Differential equation, free space electrostatics, 50–51 Differential operator: cylindrical coordinates, 8 spherical coordinates, 8–9 Diffusional dispersion: particle coagulation, Brownian motion, 429–432 solutes, microchannel flow, 275–278

INDEX

Diffusion coefficient: Brownian coagulation, field force effects, 440–448 colloidal particle deposition, Eulerian approach, 477–478 stagnation flow, 483–490 hydrodynamics of colloidal systems, 211 multicomponent systems, 186–191 molar electric conductivity, 195–198 numerical simulation, electrokinetics, cylindrical capillary model, 564–567 particle coagulation, Brownian motion, 429–432 without field force, 435–437 shear coagulation, 455–462 Dilute suspensions: Ohshima’s model, sedimentation potential, 370–381 boundary conditions, 373–374 definitions and solutions, 378–381 governing equations, 370–372 perturbation approach, 374–376 transport equations, 184 Dimensionless parameters: Brownian coagulation, field force effects, 443–448 circular charged capillary, electroosmotic flow: current flow, 263–265 double layers, 258–261 closed slit microchannel, electroosmotic flow, velocity measurements, 241–243 colloidal particle deposition: Eulerian approach, stagnation flow, 486–490 experimental verification, 516–521 flux equation, 527 interception, 473–475 trajectory equation, 527–528 colloidal suspensions rheology, 625–632 curved electric double layer, linear superposition approximation, 163–164 dilute suspensions, Ohshima’s cell model: sedimentation, 379–381 single charged sphere, sedimentation velocity, 377–378 dissimilar surfaces, overlapping planar layers, surface charge density, 150–151

681

electrophoresis: Levine–Neale cell model, 337–340 relaxation effects, 325–327 numerical simulation, electrokinetics, cylindrical capillary model, electrolyte flow, 564–567 planar electric double layer: Debye–Hückel approximation, 116–122 Gouy–Chapman analysis, 113–114 surface charge density, 124–125 slit charge microchannels, electroosmosis: electric current, 247–250 velocity flow, 237–238 Dipole moment: dielectric electrostatics, 56–62 electrostatics, dielectric medium, point charges, 99 Dirac delta function, hydrodynamics of colloidal systems, 206–212 Dirichlet condition, numerical simulation, electrokinetic phenomena, electrophoretic mobility, 593–596 Discrete phase, colloidal dispersion, 14 Disc surfaces, dielectric medium, charged and parallel discs, 96–97 Dispersion: Brownian coagulation, field force effects, 445–448 colloidal systems: discrete phase, 14 methods for, 23–24 condensation polymerization, 25 coordinates, solute dispersion, microchannel flow, 282 London–van der Waals forces, 391–393 microchannel flow, surface potentials, 269–270 solutes, microchannel flow, 274–286 convective-diffusional transport, 278–283 non-uniform flow dispersion, Taylor–Aris theory, 282–283 uniform flow dispersion, 280–282 diffusional and hydrodynamic dispersion, 275–278 slit microchannel, 283–286 Displacement current, multicomponent systems, charge conservation, 199 Dissimilar surfaces, overlapping planar layers, surface charge density, 149–151

682

INDEX

Distribution measurements, oil sand components, bitumen extraction: silica interactions, 641–644 zeta potential, 639–640 Divergence theorem: basic principles, 11–12 dielectric electrostatics, 57–62 electrostatics, boundary conditions, 64–68 linear dielectric, Maxwell stress, 69–73 multicomponent systems, binary solutions, 200–201 DLVO theory: colloidal systems, 27–28 London–van der Waals forces: colloidal interactions, 406–409 limitations, 415–420 Schulze-Hardy Rule, 409–412 verification, 412–415 oil droplet/fine particle flotation electrokinetics, 624 oil sand components, bitumen extraction, atomic force microscopy, 640–641 silica interactions, 644–645 Dorn effect, sedimentation potential, 223 velocity and, 365–370 Double layer parameter, colloidal particle deposition: dimensionless groups, flux equation, 527 Eulerian approach, stagnation flow, 487–490 trajectory equation, dimensionless groups, 528 Doublet formation, shear coagulation, 459–462 Drag coefficient, dilute suspensions, Ohshima’s cell model, 372 “Driving pressure” gradient, sedimentation potential and velocity, 366–370 Drop deformation, spherical dielectrics, Maxwell stress tensor, 90 Drug delivery, iontophoretic electrokinetics, 619–621 Dukhin number, microchannel surface conductance, 272–274 Dyadic product: Maxwell force, linear dielectric, 69–73 tensor operations, 10 Einstein equations, colloidal suspensions rheology, hard sphere model, 627–628 Electrical conduction, materials classification, 52–56

Electric current, slit charged microchannel, electroosmotic flow, 244–250 Electric double layer (EDL): charged interfaces, 105–111 Boltzmann distribution, 109–111 barometric equation, 110–111 crystal surfaces, 107 electrical potential distribution, 108–109 isomorphic substitution, 107 origins, 106–107 specific ion adsorption, 107 surface group ionization, 106 surface ion differential dissolution, sparingly soluble crystals, 106–107 circular charged capillary, electroosmotic flow: current flow, 263–265 thick layers, 257–261 thin layers, 257 colloidal particle deposition: Eulerian approach, stagnation flow, 486–490 flux equation, dimensionless groups, 527 Lagrangian approach, 500–509 random sequential adsorption model, 525–526 colloidal suspensions rheology, electroviscous effects, 628–632 curved layer geometries, 130–138 Debye–Hückel approximation: cylindrical geometry, 136–138 spherical geometry, 130–136 electrostatic interactions, 155–165 approximate solutions, 164–165 Derjaguin approximation, 157–162 linear superposition approximation, 162–164 electrophoresis: arbitrary Debye lengths, Henry’s solution, 312–322 Ohshima cell model, 342–344 single charged sphere, 297–298 boundary conditions, 301 relaxation effects, 301–302 electrostatic potential energy, 152–155 Gouy–Chapman model summary, 171–173 arbitrary electrolyte, 171–172 notations, 173 symmetrical (z:z) electrolyte, 172

INDEX

London–van der Waals forces, DLVO theory, 417–420 numerical simulation, electrokinetic phenomena: electrophoretic mobility, 589–590 Poisson–Boltzmann equation, 541–559 charged capillary particle interactions, 557–559 force calculations, 550–553 mesh generation, 549 validation of results, 554–557 planar surfaces: electric potential, 111–130 counterion analysis, high surface potentials, 128–130 Debye–Hückel approximation, 116–122 Debye thickness, symmetric electrolytes, 115–116 Gouy–Chapman analysis, 111–114 ionic concentrations, 125–128 surface charge density, 122–125 electrostatic interaction, 138–151 charged parallel plates, 144–145 dissimilar surfaces, 149–151 similar surfaces, 147–148 surface charge density, overlapping layers, 147–151 two charged surfaces, 138–144 salt rejection electrokinetics, porous media/membranes, 614–617 sedimentation potential and velocity, 365–370 slit charge microchannels, electroosmotic flow, 231–235 electric current density, 244–250 flow effectiveness, 243–244 surface conductance, 271–274 surface potential models, 165–169 indifferent electrolytes, 166–167 ionizable surfaces, 167–169 zeta potential, 169–171 Electric field strength: charged/parallel discs, dielectric medium, 95–97 conducting sphere, external electric field, 91–97 electrophoresis, circular cylinders, 354–356 free space electrostatics, 36–43 charged spherical shell, 41–43 two point charges, 38–41

683

slit charge microchannels, electroosmotic flow, electric current density, 249–250 spherical dielectrics, external electric field, 83–90 two-dimensional dielectric slab, external electric field, 77–83 Electric potential: curved electric double layer, spherical geometry, 132–136 electric double layers, interfacial charges, 108–109 electrophoretic mobility: Shilov–Zharkikh cell model, 348–352 single charged sphere, 299 electrostatics, 44–50 conducting sphere, external electric field, 91–92 dielectric medium, point charges, 97–99 spherical dielectrics, external electric field, 84–86 two-dimensional dielectric slab, external electric field, 78–80 macroscopic surfaces, zeta potential measurement, 656–659 numerical simulation, electrokinetic phenomena: capillary microchannel streaming potential, 573–577 cylindrical capillary model, 563–567 electroosmotic flow, 581–587 Poisson–Boltzmann equation, 543–546 planar electric double layer, 111–130 counterion analysis, high surface potentials, 128–130 Debye–Hückel approximation, 116–122 Debye thickness, symmetric electrolytes, 115–116 Gouy–Chapman analysis, 111–114 ionic concentrations, 125–128 surface charge density, 122–125, 123–125

684

INDEX

Electric potential (Continued) salt rejection electrokinetics, porous media/membranes, 614–617 slit charged microchannel, electroosmotic flow, 230–235 Electrochemical potentials: dilute suspensions, Ohshima’s cell model, 372 surface potential models, 166–167 Electrodialysis, sol purification, 27 Electrokinetics: applications: bitumen extraction, oil sands, 632–650 atomic force microscope technique, 640–641 bitumen-fine solids interaction, 645–650 bitumen-silica interaction, 641–645 zeta potential: distribution measurements, 639–640 oil sand components, 635–638 colloidal suspension rheology, 625–632 electroviscous effects, 628–632 hard sphere model, 626–628 historical background, 625–626 hazardous wastes, electroosmotic control, 617–619 iontophoretic drug delivery, 619–621 microfluidic/nanofluidic applications, 651–666 alternating current electrokinetics, membrane filtration, 659–666 zeta potential measurement, 653–659 oil droplet/fine particle flotation, 622–624 salt rejection, porous media and membranes, 613–617 colloid science and, 28–29 divergence and gradient theorems, 11–12 electroosmosis, 221–222 electrophoresis, 222–223 electrophoretic mobility, Ohshima cell model, 340–344 frequently used functions, 4–5 mathematical principles of, 1 microchannel flow, surface potentials, 268–270 non-equilibrium processes and Onsager relationships, 223–226

numerical simulation: basic principles, 537–538 computer tools and methods, 538–541 electrolyte flow, charged cylindrical capillary, 559–587 computational geometry and governing equations, 562–567 electroosmosis, axial pressure and potential gradients, 581–587 mesh generation and numerical solution, 567–570 microchannel streaming potential, 570–577 transient analysis, microchannel transport, 577–581 electrophoretic mobility, 587–605 Henry’s function, 596–597 mesh generation and numerical solution, 598–602 perturbation equations, 597–598 representative results, 602–605 spherical particles, 590–598 Poisson–Boltzmann equation, 541–559 charged capillary particles, EDL interaction, 557–559 EDL force calculation, 550–553 finite element formulation, 546–547 mesh generation, 547–549 results validation, 553–557 solution methodology, 549–550 symmetric electrolytes, 543–546 physical constants and conversion factors, 3–4 sedimentation potential, 223 Stokes theorem, 12 streaming potential, 222 tensor operations, 9–11 units, 2–3 vector and tensor integral theorems, 11–12 vector operations, 6–9 Cartesian coordinates, 7–8 cylindrical coordinates, 8 spherical coordinates, 8–9 Electrolytes: charged cylindrical capillary flow, numerical simulation, electrokinetic phenomena, 559–587 computational geometry and governing equations, 562–567 electroosmosis, axial pressure and potential gradients, 581–587

INDEX

mesh generation and numerical solution, 567–570 microchannel streaming potential, 570–577 transient analysis, microchannel transport, 577–581 colloidal particle deposition, experimental verification, 513–521 curved electric double layer: cylindrical geometry, 136–138 spherical geometry, 132–136 dilute suspensions, Ohshima model, 371–381 sedimentation potentials, 378–381 electrophoresis: arbitrary Debye lengths, Henry’s solution, 312–322 relaxation effects, 326–327 spherical particle surfaces, 304–306 electrophoretic mobility, Shilov-Zharkikh cell model, 347–352 Gouy–Chapman model, 171–172 indifferent, surface potential models, 166–167 London–van der Waals forces, Schulze–Hardy Rule, 410–412 mass conservation equation, 214 multicomponent systems, 188–191 binary solutions, 199–201 current density, 192–198 numerical simulation, electrokinetic phenomena: capillary microchannel transient analysis, 577–581 computer tools and methods, 541 planar electric double layer: Debye–Hückel approximation, 121–122 Gouy–Chapman analysis, 113–116 surface charge density, 122–125 sedimentation potential and velocity, 365–370 shear coagulation, 456–462 Electromagnetism: London–van der Waals forces, retardation, 402–403 Maxwell’s equations, 73–74 Electron clouds: dielectric materials, 52–56 electrophoresis, single charged sphere, 301–303

685

Electroneutrality: electrical potential distribution, electric double layers, interfacial charges, 108–109 electrophoresis, single charged sphere, 297–298 Gouy–Chapman model: arbitrary electrolyte, 171–172 symmetric electrolytes, 172 multicomponent systems, binary solutions, 199–200 numerical simulation, electrokinetic flow: cylindrical capillary model, 560–562 electrophoretic mobility, 588–590 planar electric double layer: Debye–Hückel approximation, 120–122 ion concentrations, 126–128 slit charge microchannels, electroosmotic flow, 245–250 Electroosmosis: basic principles, 221–222 hazardous waste control, 617–619 microchannel flow: circular charged capillary, 253–268 current flow, 262–265 electroviscous effect, 266–268 Helmholtz–Smoluchowski equation, thin double layers, 257 streaming potential analysis, 265–266 thick double layers, 257–261 closed slit microchannel, 240–243 effectiveness, 243–244 slit charged channel, 230–240 dispersion, 284–285 electric current, 244–250 electric potential, 230–235 flow velocity, 235–238 surface potentials, 269–270 volumetric flow rate, 238–240 numerical simulation, electrokinetic flow: axial pressure and electric potential gradients, 581–587 cylindrical capillary model, 561–562 Electroosmotic pressure, 221–222 Electrophoretic mobility: arbitrary Debye length, 308–327 alternate velocities, 322–324 Henry’s solution, 311–322

686

INDEX

Electrophoretic mobility (Continued) perturbation approach, 309–311 relaxation effects, 324–327 circular charged capillary, electroosmotic flow, double layers, 261 circular cylinders, electric field, 354–356 closed slit microchannel, electroosmotic flow, 240–243 colloidal systems, 29–30 concentrated suspension mobility, 327–353 cell model prediction accuracy, 352–353 conductivity, 344–346 hydrodynamic cell models, 328–332 Levine–Neale cell model, 333–340 Ohshima cell model, 340–344 Shilov–Zharkikh cell model, 346–352 concentrated suspensions, sedimentation potential, 381–386 defined, 306 dilute suspensions, Ohshima’s cell model, sedimentation potential, 379–381 electrokinetics, 222–223 numerical simulation, 587–605 Henry’s function, 596–597 mesh generation and numerical solution, 598–602 perturbation equations, 597–598 representative results, 602–605 spherical particles, 590–598 research background, 295–296 single charged sphere, 296–308 boundary conditions, 299–300 relaxation, 301–302 retardation, 301 surface conductance, 302–303 governing equations, 298–299 less than Debye length, 303–306 more than Debye length, 306–308 transport mechanisms, 296–298 Electrostatic forces: applications: charged and parallel discs, dielectric medium, 95–97 conducting sphere, external electric field, 91–97 electric potential and field strength, 91–92 Maxwell electrostatic stress, 94–95 surface charge density, 92–93

dielectric sphere, external electric field, 83–90 electric potential and field strength, 84–86 Maxwell electrostatic stress, dielectric sphere, 87–90 polarization surface charge density, 86–87 point charges, dielectric medium, 97–99 two-dimensional dielectric slab, external electric field, 77–83 electric potential and field strength, 78–80 Maxwell electrostatic stress, 81–83 polarization surface charge density, 80–81 boundary condition equations, 62–68 colloidal particle deposition: Eulerian approach: Brownian diffusion, 492–497 stagnation flow, 485–490 experimental verification, 515–521 trajectory equation, dimensionless groups, 527–528 colloidal particle stabilization, 21–23 colloidal suspensions rheology, 625–632 curved electric double layer, 155–165 approximate solutions, 164–165 Derjaguin approximation, 157–162 linear superposition approximation, 162–164 dielectrics, 56–62 free space basics, 33–50 electric field strength, 36–43 charged spherical shell, 41–43 two point charges, 38–41 electric potential, 44–50 charged spherical shell, 49–50 fundamental principles, 33–36 Gauss law, 43–44 free space equations, integral form, 50–51 linear dielectrics, Maxwell stress, 68–73 materials classification, 50–56 Maxwell’s electromagnetism equations, 73–74 nomenclature, 74–75 numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, 554–557 particle coagulation, 428–429 planar electric double layer, 138–151

INDEX

charged parallel plates, 144–145 dissimilar surfaces, 149–151 similar surfaces, 147–148 surface charge density, overlapping layers, 147–151 two charged surfaces, 138–144 potential energy, 152–155 shear coagulation, 455–462 Electrostatic repulsion number, colloidal particle deposition, Lagrangian approach, 503–509 Electrostriction term, Maxwell force, linear dielectric, 68–73 Electroviscous flow: circular charged capillary, electroosmotic flow, 266–268 colloidal suspensions rheology, 628–632 slit charge microchannels, 252–253 Electrowetting on a dielectric (EWOD), colloidal systems, 29–30 Emulsification, colloidal dispersion and, 24 Emulsion polymerization, condensation methods, 25 Energy equation, single-component system, 181 Equations of motion: Cartesian coordinates, 214 cylindrical coordinates, 215 spherical coordinates, 216 Equatorial trajectories, shear coagulation, 451–462 Equivalent electric conductivity, multicomponent systems, 193–198 Eulerian approach, colloidal particle deposition, 477–497 basic principles, 469–471 spherical collector: Brownian deposition with external forces, 490–497 Brownian deposition without external forces, 478–482 stagnation flow, Brownian diffusion with external forces, 482–490 Exponential integral, dilute suspensions, Ohshima model, 379–381 Extended DLVO (XDLVO), London–van der Waals forces, 420 External electrical field, electrophoresis, spherical particle surfaces, 304–306 Faraday constant, multicomponent systems, 189–191 momentum equation, 204

687

FEMLab, numerical simulation applications, 539–541 Ferric hydroxide sol, condensation methods, 25 Fick’s law: electrokinetics, Onsager reciprocity relationships, 224–226 multicomponent systems, 186–191 particle coagulation, Brownian motion, 429–432 Field force: Brownian coagulation, effects of, 437–448 Brownian coagulation without, 434–437 shear coagulation and, 451–462 Filter coefficient, colloidal particle deposition, interception, 474–475 Fine solids, oil sand components, bitumen extraction, 645–650 good processing ore, 645–647 poor processing ore, 647–648 Finite element technique, numerical simulation, electrokinetic phenomena: computer tools and methods, 539–541 cylindrical capillary model, electrolyte flow, mesh generation, 567–570 electrophoretic mobility, mesh generation, 598–605 Poisson–Boltzmann equation, 542–543, 546–547 mesh generation, 547–549 validation of results, 553–557 Finite-thickness electric double layer, electrophoresis, boundary conditions, 301–303 Flocculation: Brownian coagulation, field force effects, 445–448 London–van der Waals forces, DLVO theory, 408–409 planar electric double layer, overlapping similar surfaces, charge density, 148 shear coagulation, 460–462 Flow field verification, colloidal particle deposition, 512–521 Flow velocity: slit charge microchannels, electroosmosis, 235–238 uncharged spherical particles, sedimentation potential, 363–365 Fluid material density, Maxwell force, linear dielectric, 68–73

688

INDEX

Fluid mechanics, electrophoretic mobility, hydrodynamic cell model, 330–332 Fluid velocity field, hydrodynamics of colloidal systems, 208–211 Flux equations: colloidal particle deposition, dimensionless groups, 527 multicomponent systems, 186–191 Force calculations: numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, 550–553 planar electric double layer: charged parallel plates, 144–146 dissimilar surfaces, overlapping planar layers, 150–151 Free charge density: electrophoresis, arbitrary Debye lengths, Henry’s solution, 318–322 electrostatic conductors, 51–56 multicomponent systems, momentum equation, 203–204 numerical simulation, electrokinetic phenomena, cylindrical capillary model, 563–567 planar electric double layer, Debye–Hückel approximation, 118–122 slit charge microchannels, electroosmotic flow, 231–235 Free space electrostatics, 33–50 electric field strength, 36–43 charged spherical shell, 41–43 two point charges, 38–41 electric potential, 44–50 charged spherical shell, 49–50 fundamental principles, 33–36 Gauss law, 43–44 Fuchs derivation, particle coagulation, 429 Gaussian distribution: colloidal particle deposition, experimental verification, 516–521 particle coagulation, Brownian motion, 430–432 solute dispersion, microchannel flow, convection-diffusion, 281–283 Gaussian elimination procedure, numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, 550 Gaussian quadrature, numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, 553

Gauss law: electrostatics, 43–44 boundary conditions, 66–68 dielectric materials, 57–62 multicomponent systems, charge conservation, 198–199 Gauss theorem, curved electric double layer, spherical geometry, 135–136 Generalized minimum residual (GMRES) technique, numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, 550 Geometric properties: curved electric double layer: cylindrical geometry, 136–138 Derjaguin approximation, 158–162 spherical geometry, 130–136 dielectric medium, charged and parallel discs, 96–97 electrostatics: charged spherical shell, electric field strength, 41–43 electric field strength, 38–41 London–van der Waals forces, Hamaker constant, 397–403 Gibbs–Duhem equation, planar electric double layer, charged planar surfaces, electrostatic interaction, 141–144 Gibbs free energy change, electrostatic potential energy, 153–155 Gold sol, condensation methods, 25 Gouy–Chapman analysis: arbitrary electrolytes, 171–172 curved electric double layer, spherical geometry, 131–136 planar electric double layer, 111–114 surface charge density, 124–125 zeta potential, 169–171 Gouy–Chapman diffuse double layer model, electrical potential distribution, 108–109 Gradient theorem, basic principles, 11–12 Gravitational body force: colloidal particle deposition: dimensionless groups, flux equation, 527 Eulerian approach: spherical collectors, 495–497 stagnation flow, 485–490 Lagrangian approach, 501–509 trajectory equation, dimensionless groups, 528 colloidal particles, 17

INDEX

dilute suspensions, Ohshima’s cell model, boundary conditions, 374 Groundwater contamination, electroosmotic control, 618–619 Hamaker constant: Brownian coagulation, field force effects, 446–448 colloidal particle deposition, 17 Eulerian approach, stagnation flow, 486–490 Lagrangian approach, 509 colloidal systems, history of, 28–29 London–van der Waals forces, 393–403 cohesion work, 401–402 Derjaguin approximation, 398–400 dispersion, bodies in vacuum, 392–393 DLVO verification, 412–415 electromagnetic retardation, 402–403 intervening medium, 403–406 surface element integration, 400–401 shear-based coagulation, hydrodynamic and field forces, 459–462 Happel cell model: colloidal particle deposition: Eulerian approach, external forces, 491–497 experimental verification, 520–521 interception, 473–475 electrophoretic mobility, 330–332 Levine–Neale cell model, 333–340 uncharged spherical particles, sedimentation potential, 364–365 Hard sphere model, colloidal suspensions rheology, 626–628 Hazardous wastes, electroosmotic control, 617–619 Helmholtz free energy, electrostatic potential energy, 153–155 Helmholtz–Smoluchowski equation: circular charged capillary, electroosmotic flow, thin double layers, 257 electrophoresis: arbitrary Debye length solutions, 308–327 single charged sphere, 308 velocity equations, 322–324 macroscopic surfaces, zeta potential measurement, 655–659

689

numerical simulation, electrokinetics, cylindrical capillary model, electrolyte flow, 567 solute dispersion, microchannel flow, slit channel dispersion, 285–286 Henry’s function: dilute suspensions, Ohshima’s cell model, 380–381 electrophoresis: arbitrary Debye lengths, 309, 311–322 circular cylinders, electric field, 354–356 Levine–Neale cell model, 333–340 numerical simulation, 587, 596–597, 601–605 relaxation effects, 324–327 Hermite collocation technique, numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation validation, 554–557 Hindered settling: electrophoretic mobility, hydrodynamic cell models, 330–332 uncharged spherical particles, sedimentation potential, 364–365 Hogg–Healy–Fuerstenau (HHF) expression: curved electric double layer, Derjaguin approximation, 159–162 electrostatic potential energy, 155 numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation validation, 556–557 Hückel limit, numerical simulation, electrokinetic phenomena, electrophoretic mobility, 596 Hückel solution: electrophoresis: arbitrary Debye lengths, 308–327 arbitrary Debye lengths, Henry’s solution, 315–322 circular cylinders, electric field, 355–356 Levine–Neale cell model, 333–340 velocity equations, 322–324 electrophoresis, spherical particles, 306–308 “Hydration forces,” London–van der Waals forces, DLVO verification, 414–415 Hydrodynamics: Brownian coagulation, field force effects, 441–448

690

INDEX

Hydrodynamics (Continued) colloidal particle deposition: Eulerian approach, stagnation flow, 484–490 Lagrangian approach, 500–509, 501–509 porous media, 521–522 porous media transport, 525–526 transport models, porous media, 524–526 colloidal systems, 204–212 dilute suspensions, Ohshima’s cell model, boundary conditions, 374 electrophoresis: arbitrary Debye lengths, Henry’s solution, 314–322 cell models for, 328–332 numerical simulation, electrokinetics, electrophoretic mobility, 594–596 particle coagulation, 428–429 shear coagulation, 451–462 solute dispersion, microchannel flow, 275–278 Hydrofluoric acid, microfluid electrokinetics, 650–653 Hydrophilicity, bitumen extraction, oil sands, 634–635 Hydrophobic interactions, colloidal particle deposition, experimental verification, 516–521 Hyperbolic sine function, planar electric double layer, Debye–Hückel approximation, 116–122 Hyperfiltration, salt rejection electrokinetics, porous media and membranes, 614–617 Impinging jet geometry, colloidal particle deposition, stagnation flow, 482–490 Indifferent electrolytes, surface potential models, 166–167 Inertia, colloidal particle deposition, 475–476 Infinite dilution, multicomponent systems, molar electric conductivity, 193–198 Infinitesimal thickness, electrostatics, Maxwell stress tensor, 71–72 Infinity, numerical simulation, electrokinetic phenomena, computer tools and methods, 540–541 Insulator materials, electrostatic classification, 51–56

Insulin delivery, iontophoretic electrokinetics, 621 Integral equations: electrophoretic mobility, Levine–Neale cell model, 336–340 free space electrostatics, 50 Interception, colloidal particle deposition, 471–475 deposition efficiency, 510–512 Lagrangian approach, 504–509 Interfacial charge, electric double layer, 105–111 Boltzmann distribution, 109–111 barometric equation, 110–111 crystal surfaces, 107 electrical potential distribution, 108–109 isomorphic substitution, 107 origins, 106–107 specific ion adsorption, 107 surface group ionization, 106 surface ion differential dissolution, sparingly soluble crystals, 106–107 Internal suspension mobility, Shilov–Zharkikh cell model, 350–352 Intervening medium, London–van der Waals forces, 403–406 Ion adsorption, surface ions, electric double layers, 107 Ionic concentrations: multicomponent systems, 188–191 numerical simulation, electrokinetic phenomena, electrophoretic mobility, 588–590 planar electric double layer, 125–128 Ionic flux: dilute suspensions, Ohshima’s cell model, 371–372 Nernst–Planck equation, 213 Ionic species conservation equation, 212 Ionizable surfaces, models, 167–169 Iontophoretic drug delivery, electrokinetics, 619–621 Ion transport: electrophoresis: relaxation effects, 324–327 single charged sphere, 297–298 Debye length, 303–306 electrostatic materials, 55–56 microchannel surface conductance, 274 planar electric double layer, surface charge density, 122–125 Isomorphic substitution, surface ions, electric double layers, 107

INDEX

Kaolinite clays, oil sand components, bitumen extraction, 637–638 Kinetic equation, Brownian coagulation without field force, 436–437 Korteweig–Helmholtz electric force, linear dielectric, Maxwell stress, 68–73 Kronecker delta, tensor operations, 10 Kuwabara cell model: colloidal particle deposition, 520–521 concentrated suspensions, sedimentation potential, 381–386 electrophoretic mobility: circular cylinders, electric field, 354–356 hydrodynamic cell model, 331–332 Levine–Neale cell model, 333–340 Ohshima cell model, 340–344 uncharged spherical particles, sedimentation potential, 364–365 Lagrangian approach: colloidal particle deposition, basic principles, 470–471 numerical simulation, electrokinetic phenomena: electrophoretic mobility, 590 Poisson–Boltzmann equation, 548–549 Lamb’s solution, colloidal particle deposition, Lagrangian approach, 505–509 Laminar shear flow, coagulation, 451–462 Langevin equation, particle coagulation, Brownian motion, 431–432 Langmuirian adsorption, colloidal particle deposition, porous media transport models, 524–526 Laplace’s equation: alternating current electrokinetics, 666 dilute suspensions, Ohshima’s cell model, boundary conditions, 373–374 electrophoresis: arbitrary Debye lengths, Henry’s solution, 318–322 single charged sphere, boundary conditions, 301–303 electrostatics: boundary conditions, 62–68, 67–68 dielectric materials, 58–62 electric potential, 49–50 spherical dielectrics, 84–86

691

two-dimensional dielectric slab, external electric field, 78–80 numerical simulation, electrokinetic phenomena, electrophoretic mobility, 590–596, 602–606 Henry’s function, 596–597 mesh generation, 600–602 Laplacian operator: Cartesian coordinates, 8 numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, 547 single-component system, 180–181 solute dispersion, microchannel flow, convection-diffusion, 279–283 Latex particles, colloidal suspensions rheology, electroviscous effects, 629–632 Leaky dielectric model, Maxwell stress tensor, 90 Leibnitz’s rule, Brownian coagulation, field force effects, 439–448 Lennard–Jones potential, London–van der Waals forces: dispersion, bodies in vacuum, 392–393 Hamaker’s approach, 393–403 Levesque-type deposition rate, colloidal particle deposition, experimental verification, 515–521 Levich equation. See Lighthill–Levich equation Levine–Neale cell model: concentrated suspensions, sedimentation potential, 381–386 electrophoretic mobility: prediction accuracy, 352–353 Shilov–Zharkikh cell model vs., 349–352 electrophoretic mobility, concentrated suspensions, 333–340 Lifshitz theory, London–van der Waals forces, dispersion, bodies in vacuum, 392–393 Lighthill–Levich equation, colloidal particle deposition: Eulerian approach, 481–482 spherical collector, 493–497 stagnation flow, 489–490 experimental verification, 512–521

692

INDEX

Linear dielectrics, Maxwell stress, 68–73 Linear perturbation equations, electrophoresis, arbitrary Debye lengths, Henry’s solution, 311–322 Linear superposition approximation, curved electric double layer, electrostatic forces, 156–157, 162–165 Liquid flow, in microchannels, 229–230 Local average velocity, multicomponent systems, 182–183 London–van der Waals forces: Brownian coagulation, field force effects, 440–448 colloidal particles, 17 deposition: Lagrangian approach, 500–509 spherical collector, 493–497 DLVO theory, 406–409 Eulerian deposition, stagnation flow, 485–490 dispersion forces, bodies in vacuum, 391–393 limitations of, 415–420 verification of, 412–415 Hamaker’s approach, 393–403 cohesion work, 401–402 Derjaguin approximation, 398–400 electromagnetic retardation, 402–403 surface element integration, 400–401 intervening dielectric medium, 403–406 particle coagulation, 428 Schulz–Hardy rule, 409–412 shear coagulation, 454–462 Longitudinal diffusion, solute dispersion, microchannel flow, 277–278 Long-term deposition behavior, colloidal particle deposition, 522–523 Macroscopic model: alternating current electrokinetics, 661–666 London–van der Waals forces, dispersion, bodies in vacuum, 392–393 zeta potential measurement, 653–659 Mass center coordinate, solute dispersion, microchannel flow, convection-diffusion, 281–283 Mass conservation equation: electrolyte solution, 212 multicomponent systems, 183–184 single-component system, 180–181

Mass transfer, colloidal particle deposition: Eulerian approach, 493–497 experimental verification, 516–521 verifications, 512–521 Materials, electrostatic classification, 51–56 Maxwell’s electromagnetism equations, 73–74 Matrix equations, numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, 550 Maxwell’s equations, electromagnetism, 73–74 Maxwell stress tensor: conducting sphere, external electric field, 94–95 electrophoresis: arbitrary Debye lengths, 315–322 single charged sphere, boundary conditions, 301–303 linear dielectric materials, 68–73 numerical simulation, electrokinetic phenomena: Poisson–Boltzmann equation, charged capillary particles, EDL interaction, 558–559 Poisson–Boltzmann equation, EDL force calculations, 550–553 spherical dielectrics, 87–90 two-dimensional dielectric slab, external electric field, 81–83 Mean square displacement, particle coagulation, Brownian motion, 431–432 Membrane filtration, alternating current electrokinetics, 659–666 Mesh generation, numerical simulation, electrokinetic phenomena: cylindrical capillary model, electrolyte flow, 567–570 electrophoretic mobility, 590, 598–602 Poisson–Boltzmann equation, 547–549 Microchannel flow: electroosmotic flow: circular charged capillary, 253–268 current flow, 262–265 electroviscous effect, 266–268 Helmholtz–Smoluchowski equation, thin double layers, 257 streaming potential analysis, 265–266 thick double layers, 257–261

INDEX

closed slit microchannel, 240–243 effectiveness, 243–244 slit charged channel, 230–240 electric current, 244–250 electric potential, 230–235 flow velocity, 235–238 volumetric flow rate, 238–240 electroviscous flow, 252–254 high surface potential, 268–270 liquid flow, 229–230 numerical simulation, electrokinetic phenomena, cylindrical capillary model: computer tools and methods, 540–541 electrolyte flow, 559–587 computational geometry and governing equations, 562–567 electroosmosis, axial pressure and potential gradients, 581–587 mesh generation and numerical solution, 567–570 microchannel streaming potential, 570–577 transient analysis, microchannel transport, 577–581 Poisson–Boltzmann equation, 542–546 EDL interaction force, 557–559 solute dispersion, 274–286 convective-diffusional transport, 278–283 non-uniform flow dispersion, Taylor–Aris theory, 282–283 uniform flow dispersion, 280–282 diffusional and hydrodynamic dispersion, 275–278 slit microchannel, 283–286 streaming potential, slit channels, 251–252 surface conductance, 270–274 Micro-electromechanical systems (MEMS): electrokinetic phenomena, 29–30 electrokinetics applications, 650–653 surface potentials, 269–270 Microfluidics: electrokinetics applications, 651–666 alternating current electrokinetics, membrane filtration, 659–666 zeta potential measurement, 653–659 numerical simulation, electrokinetic flow, 559–562

693

Microscopic model, London–van der Waals forces, dispersion, bodies in vacuum, 392–393 Micro total analysis systems (µTAS), electrokinetics applications, 650–653 Migration equations, multicomponent systems, 187–191 Migration potential: electrokinetics, 223 sedimentation potential and velocity, 365–370 Minimum plate height, solute dispersion, microchannel flow: non-uniform dispersion, 282–283 slit channel dispersion, 284–285 Mobility: electrophoresis, 306 concentrated suspensions, 327–353 cell model prediction accuracy, 352–353 conductivity, 344–346 hydrodynamic cell models, 328–332 Levine–Neale cell model, 333–340 Ohshima cell model, 340–344 Shilov–Zharkikh cell model, 346–352 relaxation effects, 324–327 single charged sphere, relaxation effects, 302 multicomponent systems, 188–191 sedimentation potential and velocity, 367–370 Modified Poisson–Boltzmann (MPB) equation, slit charge microchannels, electroosmotic flow, surface potentials, 270 Molar concentration, multicomponent systems, 182–183 current density, 191–198 Molar electric conductivity, multicomponent systems, current density, 193–198 Molar flux, multicomponent systems, 188–191 Molar solute flux equation, multicomponent systems, 186–191 “Molecular condenser,” electric double layers, interfacial charges, 108–109 Momentum conservation, summary of, 214 Momentum equation: circular charged capillary, electroosmotic flow, 256–257

694

INDEX

Momentum equation (Continued) multicomponent systems, 203–204 planar electric double layer, charged planar surfaces, electrostatic interaction, 139–144 principles of, 212 Multicomponent systems: numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, 542–543 transport equations, 181–204 basic definitions, 181–183 binary electrolyte solution, 199–201 Boltzmann distribution, 201–203 conservation of change, 198–199 convection-diffusion-migration equation, 185–191 current density, 191–198 mass conservation, 183–184 momentum equations, 203–204 Multiphysics modeling, numerical simulation, computer tools and methods, 539–541 Nanofluidics, electrokinetics applications, 651–666 alternating current electrokinetics, membrane filtration, 659–666 zeta potential measurement, 653–659 Navier–Stokes equation: alternating current electrokinetics, 663–666 circular charged capillary, electroosmotic flow, 256–257 colloidal particle deposition: Brownian diffusion, convection-diffusion transport, 471 Eulerian approach, 478 electrokinetics, Onsager reciprocity relationships, 224–226 electrophoresis: arbitrary Debye lengths: Henry’s solution, 314–322 perturbation approach, 310–311 single charged sphere: boundary conditions, 300–303 Debye length, 306–308 retardation effect, 301 hydrodynamics of colloidal systems, 205–212 multicomponent systems, 202–203 momentum equation, 203–204

numerical simulation: capillary microchannel streaming potential, boundary conditions, 571–572 computer tools and methods, 538–541 cylindrical capillary model, electrolyte flow, 559–562 mesh generation, 568–570 velocity field determination, 564–567 electrokinetics applications, 537–538 electrophoretic mobility, 587–590, 592–596 Henry’s function, 597 mesh generation, 600–602 perturbation effects, 598 planar electric double layer, charged planar surfaces, electrostatic interaction, 139–144 single-component system, 180–181 slit charge microchannels, electroosmosis, 235–238 solute dispersion, microchannel flow, convection-diffusion, 279–283 Nernst–Einstein equation: ion mobility, 213 multicomponent systems, 189–191 Nernst–Planck equations: dilute suspensions, Ohshima’s cell model, 371–372 electrophoresis: arbitrary Debye lengths, perturbation approach, 310–311 single charged sphere, 297–299 boundary conditions, 300–303 relaxation effects, 302 ionic flux, 213 multicomponent systems, 190–191 Boltzmann distribution, 201–203 numerical simulation: capillary microchannel: streaming potential, 571–572 transient electrolyte transport, 578–581 computer tools and methods, 538–541 cylindrical capillary model: electrolyte flow, 559–562 ion transport, 564–567 mesh generation, 568–570 electrokinetics applications, 537–538 electroosmotic flow velocity profiles, 587

INDEX

electrophoretic mobility, 587–590, 592–596 mesh generation, 600–602 perturbation effects, 598 slit charge microchannels, electroosmotic flow, 234–235 Net radial force, spherical dielectrics, Maxwell stress tensor, 89–90 Neumann condition, numerical simulation, electrokinetic phenomena, electrophoretic mobility, 593–596 Newtonian fluids: colloidal suspensions rheology, hard sphere model, 626–628 dilute suspensions, Ohshima model, 371 uncharged spherical particles, 363–365 Newton–Raphson technique, numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation validation, 554–557 Newton’s laws of motion: colloidal particle deposition: inertial deposition, 475–476 Lagrangian approach, 470–471 particle coagulation, Brownian motion, 431–432 Nomenclature: colloidal systems, 30–31 electrostatics, 74–75 Non-conducting cylinders, electrophoresis, arbitrary Debye lengths, Henry’s solution, 321–322 Non-conducting spherical particle, electrophoresis, arbitrary Debye lengths, Henry’s solution, 313–322 Non-conjugated flow, electrokinetics, Onsager reciprocity relationships, 225–226 Non-dimensionalization: colloidal particle deposition, inertial deposition, 476 electrophoresis, velocity equations, 322–324 numerical simulation, electrokinetic phenomena, electrophoretic mobility, 594–596 Non-equilibrium processes: circular charged capillary, electroosmotic flow, current flow, 264–265 electrokinetics, 223–226

695

Nonlinearity, numerical simulation, electrokinetic phenomena, computer tools and methods, 540–541 Non-uniform dispersion, solute dispersion, microchannel flow, 282–283 Normalized excess charge, planar electric double layer, ion concentrations, 126–128 No-slip condition: hydrodynamics of colloidal systems, 205–212 multicomponent systems, momentum equation, 204 Numerical simulation, electrokinetic phenomena: basic principles, 537–538 computer tools and methods, 538–541 electrolyte flow, charged cylindrical capillary, 559–587 computational geometry and governing equations, 562–567 electroosmosis, axial pressure and potential gradients, 581–587 mesh generation and numerical solution, 567–570 microchannel streaming potential, 570–577 transient analysis, microchannel transport, 577–581 electrophoretic mobility, 587–605 Henry’s function, 596–597 mesh generation and numerical solution, 598–602 perturbation equations, 597–598 representative results, 602–605 spherical particles, 590–598 Poisson–Boltzmann equation, 541–559 charged capillary particles, EDL interaction, 557–559 EDL force calculation, 550–553 finite element formulation, 546–547 mesh generation, 547–549 results validation, 553–557 solution methodology, 549–550 symmetric electrolytes, 543–546 Ohm’s law: electrokinetics, Onsager reciprocity relationships, 224–226 electrophoretic mobility: Shilov–Zharkikh cell model, 347–352 single charged sphere, 297–298 materials classification, 52–56

696

INDEX

Ohm’s law (Continued) multicomponent systems, current density, 192–198 Ohshima cell model: concentrated suspensions, sedimentation potential, 381–386 dilute suspensions, sedimentation potential, 370–381 boundary conditions, 373–374 definitions and solutions, 378–381 governing equations, 370–372 perturbation approach, 374–376 electrophoretic mobility, 340–344 prediction accuracy, 353 Oil droplet/fine particle flotation, electrokinetics, 622–624 Oil sands: bitumen extraction, electrokinetics, 632–650 atomic force microscope technique, 640–641 bitumen-fine solids interaction, 645–650 bitumen-silica interaction, 641–645 zeta potential: distribution measurements, 639–640 oil sand components, 635–638 shear coagulation and recovery of, 461–462 Onsager reciprocity relationships: circular charged capillary, electroosmotic flow: current flow, 264–265 streaming potential, 266 concentrated suspensions, sedimentation potential, 383–386 dilute suspensions, Ohshima’s cell model, sedimentation potential, 380–381 electrokinetics, 223–226 electrophoretic mobility, cell model prediction accuracy, 352–353 sedimentation potential and velocity, 367–370 slit charge microchannels, electroosmotic flow: electric current density, 250 streaming potential, 251–252 Optical microscopy, colloidal systems, 28–29 Orthokinetic coagulation, basic principles, 248

Oseen tensor, hydrodynamics of colloidal systems, 206–212 Osmotic stress, numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, 550–553 charged capillary particles, EDL interaction, 558–559 “Outer problem,” electrophoresis, single charged sphere, boundary conditions, 301–303 Overlapping planar layers, surface charge density, 147–151 dissimilar surfaces, 149–151 similar surfaces, 147–148 Packed bed deposition, colloidal particle deposition: chemical heterogeneity, 523–524 experimental verification, 516–521 Parallel discs, dielectric medium, electrostatic properties, 95–97 Parallel plate channels, colloidal particle deposition, experimental verification, 513–521 Partial differential equations (PDEs), numerical simulation, electrokinetics: computer tools and methods, 538–541 cylindrical capillary model, mesh generation, 568–570 Poisson–Boltzmann equation, 546–547 Particle balance equation, colloidal particle deposition, interception, 474–475 Particle coagulation: Brownian motion, 429–432 Smoluchowski solution: field force effects, 437–448 without field force, 434–437 collision frequency, 432–433 dynamics of, 428–429 research background, 427–428 shear-based coagulation, 448–462 hydrodynamic and field forces, 451–462 Smoluchowski solution, absence of Brownian motion, 448–451 Particle flotation, electrokinetics, 622–624 Particle flux equation, summary, 216 Particle hydrodynamic velocity, summary of, 214 Particle number concentration, electrophoretic mobility, hydrodynamics, 328–332

INDEX

Particle size ranges: colloidal particle deposition, 469–471 deposition efficiency, 511–512 colloidal systems, 14–21 electrophoresis, spherical particles, 303–308 shear coagulation, 451–462 Particle tracking algorithms, numerical simulation, electrokinetic phenomena, electrophoretic mobility, 589–590 Particle velocity, electrophoresis, single charged sphere, 298–299 Parts integration, slit charge microchannels, electroosmotic flow, electric current density, 248–250 Peclet number: Brownian coagulation, 447–448 shear coagulation in absence of, 448–451 colloidal particle deposition: Brownian diffusion, convection-diffusion transport, 471 Eulerian approach: Brownian diffusion, 479–482 stagnation flow, 488–490 experimental verification, 515–521 spherical collectors, 494–497 colloidal suspensions rheology, 625–632 electroviscous effects, 629–632 hard sphere model, 626–628 electrophoresis, single charged sphere, 303–306 salt rejection electrokinetics, porous media/membranes, 616–617 shear coagulation, 459–462 Perikinetic coagulation, basic principles, 248 Permittivity: alternating current electrokinetics, 660–666 dielectric materials, 55–56 electrostatics, 57–62 electrophoresis, single charged sphere, relaxation effects, 301–303 multicomponent systems, momentum equation, 203–204 numerical simulation, electrokinetic phenomena: cylindrical capillary model, 563–567 electrophoretic mobility, 591–596 planar electric double layer, surface charge density, 123–125

697

two-dimensional dielectric slab, external electric field, 77–83 Perturbation theory: concentrated suspensions, sedimentation potential, 385–386 dilute suspensions, Ohshima’s cell model, 374–376 electrophoretic mobility: arbitrary Debye lengths, 309–311 numerical simulation, 587–590, 597–598 Ohshima cell model, 342–344 relaxation effects, 325–327 Shilov–Zharkikh cell model, 346–352 Physical constants, SI unit values, 3 Planar electric double layer: electric potential, 111–130 counterion analysis, high surface potentials, 128–130 Debye–Hückel approximation, 116–122 Debye thickness, symmetric electrolytes, 115–116 Gouy–Chapman analysis, 111–114 ionic concentrations, 125–128 surface charge density, 122–125 electrostatic interaction, 138–151 charged parallel plates, 144–145 dissimilar surfaces, 149–151 similar surfaces, 147–148 surface charge density, overlapping layers, 147–151 two charged surfaces, 138–144 zeta potential measurement, 654–659 Planar slabs, London–van der Waals forces, Hamaker’s approach, 393–403 Point charges, electrostatics, 34–36 dielectric materials, 59–62, 97–99 electric field strength, 36–41 electric potential, 44–50 Gauss law, 43–44 Point of zero charge (PZC): electric double layers, surface group ionization, 106 oil sand components, bitumen extraction, 637–638 Poiseuille flow: circular charged capillary, electroosmotic flow, double layers, 258–261 salt rejection electrokinetics, porous media/membranes, 615–617

698

INDEX

Poiseuille flow (Continued) slit charge microchannels, electroosmosis velocity equations, 237–238 solute dispersion, microchannel flow, non-uniform dispersion, 283 Poisson equation: alternating current electrokinetics, 666 circular charged capillary, electroosmotic flow, 254–257 curved electric double layer, spherical geometry, 134–136 dilute suspensions, Ohshima’s cell model, 372 boundary conditions, 373–374 electrophoresis: arbitrary Debye lengths, perturbation approach, 310–311 Ohshima cell model, 340–344 single charged sphere, 299 boundary conditions, 300–303 Debye length, 307–308 electrostatics: boundary conditions, 62–68 dielectric materials, 58–62 electric potential, 48–50 multicomponent systems, momentum equation, 204 numerical simulation: capillary microchannel, transient electrolyte transport, 579–581 capillary microchannel streaming potential, boundary conditions, 570–572 cylindrical capillary model, 563–567 mesh generation, 568–570 cylindrical capillary model, electrolyte flow, 559–562 electrokinetics applications, 537–538 electrophoretic mobility, 587–596 mesh generation, 600–602 perturbation effects, 598 planar electric double layer: charged planar surfaces, electrostatic interaction, 139–144 Debye–Hückel approximation, 118–122 Gouy–Chapman analysis, 112–114 overlapping similar surfaces, charge density, 147–148 surface charge density, 123–125 salt rejection electrokinetics, porous media/membranes, 614–617

slit charge microchannels, electroosmotic flow, 232–235 electric current density, 248–250 summary, 213 Poisson–Boltzmann equation: circular charged capillary, electroosmotic flow, 255–257 curved electric double layer: cylindrical geometry, 136–138 electrostatic forces, 156–157 linear superposition approximation, 163–164 spherical geometry, 131–136 Taylor expansion, 164–165 dilute suspensions, Ohshima’s cell model, perturbations, 375–376 electrophoresis: arbitrary Debye lengths, perturbation approach, 310–311 Ohshima cell model, 341–344 single charged sphere, 303–306 electrostatic potential energy, 154–155 Gouy–Chapman model, 172 London–van der Waals forces, DLVO theory, 418 macroscopic surfaces, zeta potential measurement, 656–659 multicomponent systems, 202–203 numerical simulation, electrokinetic phenomena, 541–559 basic applications, 537–538 capillary microchannel: streaming potential, 572–577 transient electrolyte transport, 579–581 charged capillary particles, EDL interaction, 557–559 computer tools and methods, 539–541 cylindrical capillary model, 561–562 mesh generation, 568–570 EDL force calculation, 550–553 electrophoretic mobility, 594–596, 602–605 Henry’s function, 596–597 mesh generation, 601–602 perturbation effects, 597–598 finite element formulation, 546–547 mesh generation, 547–549 results validation, 553–557 solution methodology, 549–550 symmetric electrolytes, 543–546

INDEX

planar electric double layer: counterion analysis, surface potentials, 128–130 Debye–Hückel approximation, 116–122 electrostatic interaction, 138–151 Gouy–Chapman analysis, 113–114 slit charge microchannels, electroosmotic flow, 233–235 electric current density, 246–250 surface potentials, 270–274 summary, 213 Polarization: alternating current electrokinetics, 661–666 electrophoresis, single charged sphere: boundary conditions, 301–303 retardation, 301 electrostatic materials, 52–56 boundary conditions, 67–68 conducting sphere, 92–93 dipole moment, 56–62 spherical dielectrics, 86–87 two-dimensional dielectric slab, external electric field, 80–81 field strength, dielectric electrostatics, 56–62 iontophoretic electrokinetics, 620–621 Polymeric dispersions, condensation methods, 25 Poly-methylmethacrylate (PMMA) plate, zeta potential measurement, 654–659 Polystyrene particles, Brownian coagulation, 444–448 Population balance equation: Brownian coagulation: field force effects, 444–448 without field force, 436–437 particle coagulation, collision frequency, 433 Porous media: colloidal particle deposition, 521–524 chemical heterogeneity, 523–524 hydrodynamic dispersion, 521–522 long-term behavior, 522–523 release mechanisms, 523 transport models, 524–526 electrophoretic mobility, Levine–Neale cell model, 338–340 salt rejection electrokinetics, 613–617 Potential determining ions, surface potential models, 167

699

Potential energy, electrostatic forces, 152–155 Pressure-driven flow, slit charged microchannel, 230–235 dispersion, 283–284 Pressure gradient: circular charged capillary, electroosmotic flow: current flow, 263–265 double layers, 259–261 electrophoresis, single charged sphere, 296–298 numerical simulation, electrokinetic phenomena: capillary microchannel streaming potential, 572–577 cylindrical capillary model, 561–562 mesh generation, 569–570 electroosmotic flow, 581–587 slit charge microchannels, electroosmotic flow, electric current density, 249–250 Primary/secondary electroviscous effects, colloidal suspensions rheology, 628–632 Probability density: colloidal particle deposition, experimental verification, 519–521 electric double layers, interfacial charges, Boltzmann distribution, 111 Processing ore, oil sand components, bitumen extraction, zeta potential measurements, 645–648 Proportionality constant: electric double layers, interfacial charges, Boltzmann distribution, 111 hydrodynamics of colloidal systems, 207–212 Pseudo-chemical reaction, London–van der Waals forces, dielectric intervening medium, 403–406 Quadrature formula, electrophoresis, arbitrary Debye lengths, Henry’s solution, 317–322 Quasi-minimal residual (QMR), numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, 550 Random sequential adsorption (RSA) model, colloidal particle deposition, porous media transport, 524–526

700

INDEX

Rationalized MKS (RMKS) units, Maxwell’s equations, electromagnetism, 73–74 Reference chemical potential, electrophoretic mobility, Ohshima cell model, 340–344 Reference potential, multicomponent systems, Boltzmann distribution, 202–203 Relative viscosity, colloidal suspensions rheology, electroviscous effects, 629–632 Relaxation effects: electrophoresis: arbitrary Debye lengths, Henry’s solution, 312–322 single charged sphere, 301–303 solutions for, 324–327 sedimentation potential and velocity, 365–370 Release mechanisms, colloidal particle deposition, 523 porous media transport, 526 Repulsive force: colloidal particle deposition: Eulerian approach, stagnation flow, 486–490 experimental verification, 515–521 Lagrangian approach, 503–509 London–van der Waals forces, DLVO theory, 406–409 oil sand components, bitumen extraction, fine solids interaction, 648–650 shear coagulation, 456–462 Retardation: Brownian coagulation, field force effects, 441–448 colloidal particle deposition, dimensionless groups, flux equation, 527 electrophoresis, single charged sphere, 301 London–van der Waals forces, Hamaker constant, 402–403 Reverse osmosis: numerical simulation, electrokinetic flow, cylindrical capillary model, 560–562 salt rejection electrokinetics, porous media and membranes, 614–617 Reynolds number: colloidal particle deposition:

Brownian diffusion, convection-diffusion transport, 471 Eulerian approach: Brownian diffusion, 479–482 stagnation flow, 483–490, 488–490 inertial deposition, 475–476 interception, 473–475 dilute suspensions, Ohshima’s cell model, 371–372 electrophoresis, single charged sphere, 299 electrophoretic mobility, hydrodynamic cell models, 329–332 momentum equation, 212 uncharged spherical particles, sedimentation potential, 363–364 Rheology of colloidal suspensions, 625–632 electroviscous effects, 628–632 hard sphere model, 626–628 historical background, 625–626 Ripening process, colloidal particle deposition, long-term behavior, 522–523 Root mean square analysis, alternating current electrokinetics, 662–666 Salt rejection electrokinetics, salt rejection, porous media and membranes, 613–617 Sand grains, bitumen liberation from, 641–645 Scalar function: electrostatics, electric potential, 45–50 theorems for, 11–12 Scaled electric potential, numerical simulation, electrokinetic phenomena: capillary microchannel, transient results, 579–581 electroosmotic flow, 583–587 electrophoretic mobility, 593–596 Poisson–Boltzmann equation, 543–546 Scaled pressure, numerical simulation, capillary microchannel streaming potential, 572 Scale factors: concentrated suspensions, sedimentation potential, 385–386 SI units, 2–3 Scavenging process, shear coagulation and, 461–462

INDEX

Schmidt numbers, colloidal particle deposition: Brownian diffusion, convection-diffusion transport, 471 Eulerian approach: Brownian diffusion, 481–482 stagnation flow, 482–490 Schulze–Hardy Rule, London–van der Waals forces, 409–412 Second order tensors, basic properties, 9–10 Sedimentation potential: colloidal particle deposition, experimental verification, 518–521 concentrated suspensions, 381–386 dilute suspensions, Ohshima’s model, 370–381 boundary conditions, 373–374 definitions and solutions, 378–381 governing equations, 370–372 perturbation approach, 374–376 electrokinetics, 223 electrophoretic mobility, hydrodynamic cell models, 330–332 uncharged spherical particles, 363–365 velocity and, 365–370 Separation distance, colloidal particle stabilization, 22–23 Shape functions, numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, 547–549 Shear-based coagulation, 448–462 hydrodynamic and field forces, 451–462 Smoluchowski solution, absence of Brownian motion, 448–451 Shear flow, hydrodynamics of colloidal systems, 210–212 Shear rate, colloidal suspensions rheology, 625–632 Sherwood number, colloidal particle deposition: Brownian diffusion, convection-diffusion transport, 471 deposition efficiency, 509–512 Eulerian approach: Brownian diffusion, 481–482 external forces, 491–497 stagnation flow, 488–490 experimental verification, 512–521 long-term behavior, 522–523

701

Shilov-Zharkikh cell model: concentrated suspensions, sedimentation potential, 381–386 electrophoretic mobility, 346–352 prediction accuracy, 352–353 Silica-bitumen interaction, oil sand components, bitumen extraction, 641–645 Silver bromide sol, condensation methods, 25 Similar surfaces, overlapping planar layers, surface charge density, 147–148 Sine theorem, electrostatics, electric field strength, 40–41 Single charged sphere: dilute suspensions, Ohshima’s cell model, sedimentation velocity, 376–378 electrophoresis, 296–308 boundary conditions, 299–303 relaxation, 301–302 retardation, 301 surface conductance, 302–303 governing equations, 298–299 less than Debye length, 303–306 more than Debye length, 306–308 transport mechanisms, 296–298 Single-component system, transport equations, 180–181 Skin impedance, iontophoretic electrokinetics, 620–621 Slit charged microchannel: colloidal particle deposition, experimental verification, 515–521 dispersion in, 283–285 electroosmotic flow, 230–240 closed slit microchannel, 240–243 electric current, 244–250 electric potential, 230–235 flow velocity, 235–238 macroscopic surfaces, zeta potential measurement, 655–659 streaming potential, 251–252 Smoluchowski equation: Brownian coagulation: field force effects, 437–448 no field force, 434–437 colloidal systems, 28–29 electrophoresis, single charged sphere, 308 electrophoretic mobility: cell model prediction accuracy, 352–353

702

INDEX

Smoluchowski equation (Continued) Levine–Neale cell model, 336–340 microchannel flow, surface conductance, 270–274 sedimentation potential and velocity, 367–370 shear-based coagulation, 448–451 hydrodynamic and field forces, 459–462 Sols: condensation methods, 24–27 purification of, 26–27 Solute dispersion, microchannel flow, 274–286 convective-diffusional transport, 278–283 non-uniform flow dispersion, Taylor–Aris theory, 282–283 uniform flow dispersion, 280–282 diffusional and hydrodynamic dispersion, 275–278 slit microchannel, 283–286 Space charge density: numerical simulation, electrokinetic flow, cylindrical capillary model, 560–562 planar electric double layer, Gouy–Chapman analysis, 112–114 Sparingly soluble crystals, surface ions, electric double layers, differential dissolution, 106–107 Spatial gradients, numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, mesh generation, 547–549 Speed of light, Maxwell’s electromagnetism equations, 74 Spherical collectors, colloidal particle deposition: Eulerian approach, Brownian diffusion, 478–482, 490–497 interception, 473–475 Lagrangian approach, 497–509 Sherwood number and efficiency of, 509–512 Spherical coordinates: basic components, 8–9 colloidal particle deposition, interception, 472–475 curved electric double layer, Debye–Hückel approximation, 130–136

electrophoretic mobility, Ohshima cell model, 340–344 equations of motion, 216 single charged sphere, electrophoresis, 296–308 boundary conditions, 299–303 relaxation, 301–302 retardation, 301 surface conductance, 302–303 governing equations, 298–299 less than Debye length, 303–306 more than Debye length, 306–308 transport mechanisms, 296–298 Spherical dielectrics, external electric field, 83–90 conducting sphere, 91–97 electric potential and field strength, 84–86 Maxwell electrostatic stress, dielectric sphere, 87–90 polarization surface charge density, 86–87 Spherical particles: Brownian coagulation, Smoluchowski solution, without field force, 434–437 coagulation dynamics, 429 colloidal particle deposition: Eulerian approach, 477–478 Lagrangian approach, 498–509 electrophoresis, single charged sphere, 296–308 boundary conditions, 299–303 relaxation, 301–302 retardation, 301 surface conductance, 302–303 governing equations, 298–299 less than Debye length, 303–306 more than Debye length, 306–308 transport mechanisms, 296–298 electrostatics: dielectric materials, 59–62 electric field strength, 41–43 hydrodynamics of colloidal systems, fluid velocity field, 208–212 numerical simulation, electrokinetic phenomena: electrophoretic mobility, 590–596 mesh generation, 598–602 Poisson–Boltzmann equation, 541–559, 553 EDL interaction, charged capillary particles, 557–559

INDEX

Ohshima’s cell model: dilute suspensions, 371–381 electrophoretic mobility, 340–344 single charged sphere, sedimentation velocity, 376–378 sedimentation potential, uncharged particles, 363–365 shear coagulation, 448–451 hydrodynamic and field forces, 451–462 Stability ratio: Brownian coagulation, field force effects, 440–448 colloidal particle deposition, 469–470 shear-based coagulation, hydrodynamic and field forces, 460–462 Stabilization, colloidal particles, 21–23 Stagnation flow, colloidal particle deposition: Eulerian approach, 482–490 experimental verification, 516–521 Lagrangian approach, 499–509 Stationary electrolyte reservoir, multicomponent systems, 202–203 Stationary surface: circular charged capillary, electroosmotic flow, double layers, 261 closed slit microchannel, electroosmotic flow, 242–243 Steady state equations: Brownian coagulation: field force effects, 438–448 without field force, 435–437 dilute suspensions, Ohshima’s cell model, 371–372 electrophoresis, single charged sphere, 298–299 numerical simulation, electrokinetic phenomena: capillary microchannel streaming potential, 572–577 cylindrical capillary model, 561–562 mesh generation, 568–570 electroosmotic flow velocity profiles, 583–587 electrophoretic mobility, 589–590 Stern plane/layer: microchannel flow, surface conductance, 270–274 zeta potential, 170–171

703

Stern potential, colloidal particle deposition, experimental verification, 516–521 Stokes–Einstein equation: Brownian coagulation: field force effects, 439–448 without field force, 436–437 colloidal particle deposition: Brownian diffusion, convection-diffusion transport, 471 deposition efficiency, 510–512 Eulerian approach, 478 stagnation flow, 484–490 porous media, 522 colloidal suspensions rheology, 626–632 electrophoretic mobility, suspension conductivity, 344–346 ion mobility, 213–214 particle coagulation, Brownian motion, 432 Stokes equations: colloidal particle deposition: Eulerian approach: Brownian diffusion, 479–482 stagnation flow, 483–490 inertial deposition, 476 Lagrangian approach, 498–509 dilute suspensions, Ohshima’s cell model, single charged sphere, sedimentation velocity, 377–378 electrophoretic mobility: hydrodynamic cell models, 329–332 Levine–Neale cell model, 335–340 Ohshima cell model, 341–344 hydrodynamics of colloidal systems, 206–212 numerical simulation, electrophoretic mobility, 587–590 particle coagulation, Brownian motion, 431–432 sedimentation potential and velocity, 366–370 uncharged spherical particles, 363–365 shear coagulation, 454–462 Stokes hydrodynamic drag force, electrophoresis, spherical particle surfaces, 305–306 Stokes theorem: basic principles, 12 electrostatics, boundary conditions, 63–68

704

INDEX

Stratum corneum electrical properties, iontophoretic electrokinetics, 620–621 Streaming potential: circular charged capillary, electroosmotic flow, 265–266 electrokinetics, 222 macroscopic surfaces, zeta potential measurement, 656–659 microchannel flow, surface conductance, 270–274 numerical simulation, electrokinetic phenomena: capillary microchannel, 570–577 boundary conditions, 570–572 numerical vs. analytical results, 572–577 cylindrical capillary model, electrolyte flow, 559–562 sedimentation potential and velocity, “driving pressure” gradient, 366 slit charged microchannel, 251–252 Stress tensor: conducting sphere, external electric field, 94–95 linear dielectric, Maxwell stress, 68–73 numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, EDL force calculations, 551–553 planar electric double layer, charged parallel plates, 145–146 spherical dielectrics, 87–90 two-dimensional dielectric slab, external electric field, 81–83 Sulphur sol, condensation methods, 25 Superposition principle: colloidal particle deposition, Lagrangian approach, 499–509 curved electric double layer, electrostatic forces, 156–157 electrophoresis: arbitrary Debye lengths, Henry’s solution, 318–322 single charged sphere, 297–298 electrostatics, 35–36 dielectric medium, point charges, 98–99 electric potential, 48–50 numerical simulation, electrokinetic phenomena, electroosmotic flow, 582–587

Surface area to volume ratio, colloidal particles, 16–21 Surface charge density: curved electric double layer, spherical geometry, 133–136 electrophoresis, arbitrary Debye lengths, 315–322 electrostatic potential energy, 154–155 electrostatics: boundary conditions, 65–68 conducting sphere, 92–93 spherical dielectrics, 86–87 two-dimensional dielectric slab, external electric field, 80–81 numerical simulation: capillary microchannel electrolyte transport, 578–581 electroosmotic flow, 582–587 oil sand components, bitumen extraction, 636–638 planar electric double layer, 122–125 overlapping layers, 147–151 dissimilar surfaces, 149–151 similar surfaces, 147–148 salt rejection electrokinetics, porous media/membranes, 616–617 Surface conductance: electrophoresis, single charged sphere, 302–303 microchannel flow, 270–274 Surface element integration (SEI): curved electric double layer, 165 London–van der Waals forces, Hamaker constant, 398, 400–401 Surface forces, colloidal particles, 16–21 Eulerian deposition, Brownian diffusion, 480–482 Surface ionization: electric double layers, 106 differential dissolution, sparingly soluble crystals, 106–107 models, 167–169 “Surface molecules,” colloidal particles, 18–21 Surface potentials: electric double layer models, 165–169 indifferent electrolytes, 166–167 ionized surfaces, 167–169 electrophoresis, relaxation effects, 325–327 microchannel flow, 268–270

INDEX

numerical simulation, electrokinetic phenomena, electrophoretic mobility, 602–605 planar electric double layer: charged planar surfaces, electrostatic interaction, 143–144 counterion analysis, 128–130 Debye–Hückel approximation, 117–122 Surface tension, colloidal particles, 17 Suspensions: colloidal suspension rheology, 625–632 electroviscous effects, 628–632 hard sphere model, 626–628 historical background, 625–626 conductivity: dilute suspensions, Ohshima’s model, sedimentation potential, 370–381 electrophoretic mobility, 344–346 uncharged spherical particles, sedimentation potential, 363–365 polymerization, condensation methods, 26 Symmetric electrolytes. See also Asymmetric electrolytes curved electric double layer: electrostatic forces, 156–157 spherical geometry, 132–136 Gouy–Chapman model, 172 numerical simulation, electrokinetic phenomena: cylindrical capillary model, 563–567 electroosmotic flow velocity profiles, 586–587 electrophoretic mobility, 594–596 Poisson–Boltzmann equation, 543–546 stress calculations, 551–553 planar electric double layer: Debye–Hückel approximation, 121–122 Gouy–Chapman analysis, 113–115 ionic concentrations, 126–128 slit charge microchannels, electroosmotic flow, 231–235 electric current density, 245–250 Symmetric tensors, basic properties, 9–10 Système International d’Unités (SI): derived quantities, 2 electrokinetic transport and, 1 physical constants, 3 scale factors, 203

705

Tangential flow filtration, alternating current electrokinetics, 662–666 Tangential shear stress, electrophoretic mobility, Happel cell model, 330–332 Taylor–Aris theory, non-uniform dispersion, solute dispersion, microchannel flow, 282–283 Taylor series expansion: curved electric double layer, Poisson–Boltzmann equation, 164–165 electrophoretic mobility, Ohshima cell model, 341–344 slit charge microchannels, electroosmotic flow, 235 electric current density, 246–250 Tensor operations: basic components, 9–11 colloidal particle deposition, Eulerian approach, 477–478 Maxwell stress tensor, linear dielectric materials, 68–73 theorems for, 11–12 Tensor product, vector operations, 7 Thermal motion, solute dispersion, microchannel flow, 276–278 Thermodynamic standpoint, electrostatic potential energy, 153–155 Trajectory equation: alternating current electrokinetics, 663–666 colloidal particle deposition, dimensionless groups, 527–528 oil droplet/fine particle flotation electrokinetics, 624 Transient electrokinetic flow, numerical simulation, capillary microchannel electrolyte transport, 577–581 Transport equations: colloidal hydrodynamics, 205–212 colloidal models, porous media, 524–526 colloidal particle deposition, Eulerian approach, stagnation flow, 489–490 electrophoresis, arbitrary Debye lengths, Henry’s solution, 312–322 governing equations, 212–216 multicomponent systems, 181–204 basic definitions, 181–183 binary electrolyte solution, 199–201 Boltzmann distribution, 201–203 conservation of change, 198–199

706

INDEX

Transport equations (Continued) convection-diffusion-migration equation, 185–191 current density, 191–198 mass conservation, 183–184 momentum equations, 203–204 overview of, 179–180 single-component system, 180–181 Transport mechanisms: electrophoresis, single charged sphere, 296–298 salt rejection electrokinetics, porous media and membranes, 613–617 Transverse diffusion, solute dispersion, microchannel flow, 277–278 Two-dimensional dielectric slab, external electric field, electrostatics, 77–83 electric potential and field strength, 78–80 Maxwell electrostatic stress, 81–83 polarization surface charge density, 80–81 Ultrafiltration: alternating current electrokinetics, 662–666 numerical simulation, electrokinetic flow, cylindrical capillary model, 560–562 sol purification, 26–27 Ultrasonic vibrations, colloidal dispersion and, 24 Uniform electric field, spherical dielectrics, 84–86 “Uniform suction” trajectory, alternating current electrokinetics, 664–666 Unit dyad, tensor operations, 10 Unretarded interactions, London–van der Waals forces, intervening medium, 403–406 Validation, numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, 553–557 Van der Waals interactions. See also London–van der Waals forces Brownian coagulation, field force effects, 446–448 coagulation dynamics, 428–429 colloidal particle deposition: deposition efficiency, 511–512

Eulerian approach: Brownian diffusion, 492–497 stagnation flow, 486–490 colloidal systems, 27–28 London–van der Waals forces, Hamaker constant, 397–403 shear coagulation, 454–462 Vector function, 6–9 Cartesian coordinates, 7–8 cylindrical coordinates, 8 electrostatics: electric field strength, 37–41 electric potential, 45–50 spherical coordinates, 8–9 Velocity profiles: alternating current electrokinetics, 663–666 circular charged capillary, electroosmotic flow, double layers, 261 colloidal particle deposition: Eulerian approach, 477–478 Brownian diffusion, 479–482 Lagrangian approach, 498–509 dilute suspensions, Ohshima model, 370–381 boundary conditions, 373–374 definitions and solutions, 378–381 governing equations, 370–372 perturbation approach, 374–376 electrophoresis: alternate forms, 322–324 spherical particle surfaces, 305–306 hydrodynamics of colloidal systems, 205–212 macroscopic surfaces, zeta potential measurement, 656–659 multicomponent systems, 182–183 numerical simulation, electrokinetic phenomena: capillary microchannel streaming potential, 573–577 electroosmotic flow, 582–587 electrophoretic mobility, 588–590 sedimentation potential and, 365–370 slit charge microchannels, electroosmotic flow, 235–238 solute dispersion, microchannel flow, mass center velocity, 282 Very large surface potentials, planar electric double layer, charged planar surfaces, electrostatic interaction, 143–144

INDEX

Very small surface potentials, planar electric double layer, charged planar surfaces, electrostatic interaction, 143–144 Viscous friction forces: Brownian coagulation, 443–448 colloidal suspensions rheology, 625–632 electroviscous flow, 628–632 hard sphere model, 626–628 Volume element, electrostatics, Maxwell stress tensor, 71–72 Volume fraction, colloidal suspensions rheology, electroviscous effects, 631–362 Volumetric charge density: dielectric electrostatics, 57–62 electrophoresis, single charged sphere, 297–298 retardation effect, 301 Volumetric flow rate: circular charged capillary, electroosmotic flow: current flow, 264–265 double layers, 258–261 electroviscous flow, 266–268 closed slit microchannel, electroosmotic flow, 241–243 slit charge microchannels, electroosmosis, 238–240 Volumetric flow ratio (VFR), electroosmotic flow effectiveness, 243–244 Water drainage, electroosmotic control, 618–619 “Water-wet” oil sands, bitumen extraction, 634–635 x-directional momentum equation: Brownian coagulation without field force, 435–437

колхоз 7/11/06

707

electrophoresis, arbitrary Debye lengths, Henry’s solution, 312–322 single-component system, 181 slit charge microchannels, electroosmosis, 235–238 y-directional momentum equation: colloidal particle deposition, Lagrangian approach, 501–509 single-component system, 181 slit charge microchannels, electroosmosis, 235–238 Zeta potential: colloidal particle deposition, experimental verification, 513–521 colloidal suspensions rheology, electroviscous effects, 628–632 dilute suspensions, Ohshima model, sedimentation velocity, 378–381 electric double layer, 169–171 electrophoresis: arbitrary Debye lengths, perturbation approach, 311 mobility, Levine–Neale cell model, 334–340 velocity equations, 322–324 macroscopic surfaces measurement, 653–659 numerical simulation, electrokinetic phenomena, Poisson–Boltzmann equation, 542 oil droplet/fine particle flotation electrokinetics, 622–624 oil sand components, bitumen extraction, 635–638 distribution measurements, 639–640 fine solids interaction, 645–648 silica interactions, 641–644 sedimentation potential and velocity, 366–370